Plane Trees and Reduction Rules

Nelson Y. Li1, Toufik Mansour2 and Augustine O. Munagi3

1The High School Affiliated to Renmin University of China, Renmin University

of China, Beijing 100080, P.R. China

2Department of Mathematics, University of Haifa, Haifa 31905, Israel2Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R.

China

3School of Mathematics, University of the Witwatersrand, WITS 2050, South

Africa

1nelsonli@eyou.com, 2toufik@math.haifa.ac.il, 3augustine.munagi@wits.ac.za

Abstract

Inspired by the results of Gelfand, Graev and Postnikov (1997) on thereduction rule for binary trees, we present several bijections on differenttypes of reduction rules for plane trees such as symmetric k-ary trees, locallysymmetric k-ary trees, symmetric even trees, symmetric noncrossing treesand Motzkin trees. We give the corresponding enumeration results, andprove the equidistribution of some classes of trees.

Keywords: Reduction rules; Ternary trees; Even trees; Noncrossing trees;k-ary trees; Catalan numbers; Motzkin numbers.AMS Classifications: 05A15, 05C05.

1 Introduction

In 2002, Postnikov ([15, Exerc. 6.C6],[12]) proposed the following problemoriginally motivated by the Chan-Robbins-Yeun conjecture on the volumeof a certain polytope [1, 18]. Start with a monomial x = x12x23 · · ·xn(n+1)

and repeatedly apply the reduction rule

xijxjk → xikxij + xjkxik for i < j < k (1.1)

until no further reduction is possible. A monomial which admits no furtherreduction under a given rule will also be called a final term. For example,if x = x12x23x24, application of rule (1.1) gives

x12x23x24 → x13x12x24+x23x13x24 → x13x14x12 +x13x24x14 +x23x13x24.

Gelfand, Graev and Postnikov [6] have shown that the number of finalterms is the n-th Catalan number Cn = 1

n+1

(

2nn

)

. The reduction rule for

1

a monomial x =∏

(i,j)∈I xij can be represented by marking n = |{i, j |

(i, j) ∈ I}| points on a horizontal line which are labelled in increasingorder, and connecting them by |I| arcs, such that each arc connects andlies above two points. This is known as the linear representation.

−→ +i j k i j k i j k

Figure 1: Linear representation of (1.1) for xijxjk.

For example, (1.1) is represented by Figure 1. Denote the set of allthe linear representations of the final terms by N (x) and its cardinality byN(x) = |N (x)|.

Postnikov and Stanley [15] have noted that for many rules, the resultingpolynomial depends on the order in which reductions are executed. This isis due to the fact that, for a given vertex i, the arc (i, j), i < j, is generallynot unique. As an illustration, fix the reduction rule

xijxjkxkh → xihxijxjk + xihxjkxkh for i < j < k < h, (1.2)

having the linear representation in Figure 2. When n = 5 one possible se-

−→ +i j k h i j k h i j k h

Figure 2: Linear representation of the reduction rule for xijxjkxkh.

quence of reductions (with the pairs of transformable variables in boldface)is given by

x12x23x34x45x56

→ x12x14x23x45x56 + x14x23x34x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x23x25x34x56

+x14x25x34x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x23x25x34x56

+x14x25x34x36x45 + x14x25x36x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x23x25x34x56

+x14x25x34x36x45 + x14x16x25x36x45 + x16x25x36x45x56,

which gives 6 final terms. But repeating the reduction by always employing

2

the longest arcs, from left to right, we obtain

x12x23x34x45x56

→ x12x14x23x45x56 + x14x23x34x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x16x23x34x45+x16x23x34x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x16x23x25x34

+x14x16x25x34x45 + x16x23x25x34x56 + x16x25x34x45x56

→ x12x14x16x23x45 + x12x16x23x45x56 + x14x16x23x25x34

+x14x16x25x34x45 + x16x23x25x34x56 + x16x25x34x36x45

+x16x25x36x45x56,

which results in 7 final terms. Henceforth, we adopt the convention torepresent reduction rules by always applying the longest possible outgo-ing arc at each vertex, from left to right. Then the terms in the setN (x12x23 · · ·xn n+1) of rule (1.1) are noncrossing alternating trees [16, Exe.6.19p] (or standard trees [6]). It is known that the number of noncrossingalternating trees with n edges is given by the n-th Catalan number Cn

(see [15, 16]). For example, all noncrossing alternating trees for n = 3 areshown in Figure 3.

1 1 1 1 12 2 2 2 23 3 3 3 34 4 4 4 4

Figure 3: Noncrossing alternating trees with 3 edges.

The purpose of this paper is to investigate several kinds of plane treesgenerated by different types of reduction rules.

A k-ary tree is a plane tree in which each vertex has outdegree 0 ork. We say that a k-ary tree T is locally symmetric if the k − 1 rightmostchildren of wi have the same induced subtrees, where wi is the leftmostchild of wi−1 and w1 is the root of T , as illustrated in Figure 4. In thecase k = 3, T is called a locally symmetric ternary tree. Let T (x) and S(x)

Ti Ti · · · Ti

T2 T2 · · · T2

T1 T1 · · · T1

wi

w2

w1

w3

Figure 4: Locally symmetric k-ary trees.

denote the generating functions for the numbers of k-ary trees and locally

3

symmetric k-ary trees, respectively. Then using the symbolic method [13,Sec. 3.9], we immediately obtain the equations: T (x) = 1 + xT k(x) andS(x) =

∑

i≥0 xiT i(xk−1), where x marks the number of internal vertices

(i.e., vertices with nonzero outdegree), the term T (xk−1) corresponds tok−1 copies of the same tree, and the index i to the length of the left branch.Applying the Lagrange inversion formula gives S(x) =

∑

n≥0 sknx

n, where

skn =

ℓ+ 1

m

(

km+ ℓ

m− 1

)

, (1.3)

with n = m(k − 1) + ℓ and 0 ≤ ℓ ≤ k − 2.This paper is organized as follows. In the next section we analyse (1.2)

and relate the final terms to locally symmetric ternary trees by establishinga bijection. In Section 3, we employ the most general form of previous rulesto obtain a relation with locally symmetric k-ary trees. In Section 4, ourtechnique is used to derive the equidistribution of symmetric even trees,symmetric ternary trees and symmetric noncrossing trees. Lastly, in Sec-tion 5 we present other reduction rules which allow us to state enumerationresults on Motzkin trees and plane trees with loops.

2 Reduction rule for locally symmetric

ternary trees

In this section we give a bijection between the final terms arising from (1.2)and locally symmetric ternary trees. Let N 3(x) denote the set of the finalterms with N3(x) = |N 3(x)|. An arc in a linear representation is said tobe visible if its vertical height from the horizontal line is maximal. Forinstance, the longest arc (1, 8) is visible in Figure 7a.

The following properties are immediate for the final term of

x = x12x23x34 · · ·xn n+1

following repeated applications of rule (1.2).(1) The linear representation is a connected graph with n+ 1 vertices,

lying on the x-axis, and n edges such that no edge passes below the x-axis.(2) If n is odd, there is only one visible arc, the longest. If n is even,

n > 0, there are two visible arcs. Moveover, the deletion of the two visiblearcs, returns a (smaller) final term with two new visible arcs. This is thethe elimination algorithm.

(3) There are four patterns (or arc-shapes) arising from an inductiveexecution of the elimination algorithm namely (a), (b), (c) and (d) in Fig-ure 5. The pattern (a) is for the first-level execution, when n is odd. Here

4

P1, P2 may be empty but they disjoint, where P1 is a final term starting at1 and P2 is a final term ending at k+1. The other patterns are analogouslyinterpreted. Note that P1 can be empty in (b) or (c), while P2 is nonemptyin (c). In (d), P1 and P2 can be empty while P3 is nonempty.

P1 P21 k + 1

(a)

←→

v

τ(a)

P1 P2 P2

P1 k k + 1

(b)

←→

v

τ(b)

Single:

P1

v

τ(b)

Multi:

P1

v′

P1

P1 P2 k k + 1

(c)

←→

v

τ(c)

Single:

P1 P2 P2

v

τ(c)

Multi:

P1 P2

v′

P1 P2

P1 P2 P3

k

k + 1

(d)

←→

v

τ(d)

Single:

P1 P2 P2

P3 P3

v

τ(d)

Multi:

P1 P2 P3

v′

P1 P2 P3

Figure 5: Final terms and locally symmetric ternary trees.

A final term y is said to be closed if its pattern is one of (a), (b),(c) and (d), and every successive application of the elimination algorithm,beginning with y, gives a final term with one of the four patterns, excludingthe empty term.

Lemma 2.1 Each final term deduced from rule (1.2) is closed.

Proof. We proceed by induction on n. When n = 0, the only final term isa vertex. For n = 1, 2, they are x12, x12x23 respectively. When n = 3, we

5

have x14x12x23 and x14x23x34, and n = 4 gives x14x12x23x45, x14x23x34x25,and x14x25x34x45. It is obvious that all of the terms are closed. We as-sume n ≥ 5, and all the final terms with n arcs obtained from (1.2) areclosed. Given the monomial x12x23x34x45 · · ·xn+1 n+2, proceed from left toright to obtain x12x23x14x45 · · ·xn+1 n+2 and x14x23x34 · · ·xn+1 n+2. Notethat all the final terms deduced from x14x45 · · ·xn+1 n+2 are closed by theinductive hypothesis. Thus inserting the part x12x23 in each term, pre-serves closure. Also with x14x23x34 · · ·xn+1 n+2, the final terms deducedfrom x14x45 · · ·xn+1 n+2 are closed by hypothesis. Now, insert x23x34 afterthe four patterns. Then only two admissible cases arise for the final terms,illustrated in Figure 6.

P21 2 3 4 k k + 1P2 P31 2 3 4

k

k + 1

Figure 6: The cases of final terms (proof of Lemma 2.1).

Applying rule (1.2) on the thick (longest) arcs in the figure shows theresulting structures are closed. Hence the proof follows by induction.

It is easy to observe that the four patterns are pairwise distinguishable.So it is possible to obtain an explicit formula for N3(x12x23 · · ·xn n+1) bycombinatorial considerations.

Given a final term with n arcs, there corresponds a unique locally sym-metric ternary tree with n internal vertices. We give a constructive demon-stration of this relation by using the elimination algorithm to define a bi-jection τ . First draw the root, that is, a vertex at the top level. Then applythe algorithm inductively, each time adding three or six edges dependingon whether one or two visible arcs (respectively) are eliminated. For thecorresponding locally symmetric ternary trees, define a(Pi), 1 ≤ i ≤ 3,as the active set of Pi containing prospective roots of nonempty subtrees.Clearly |a(Pi)| equals 1 or 2, corresponding to the single and multi casesrespectively, as illustrated in Figure 5. It is easy to see that the first elim-ination when n is odd only corresponds to the single case. Subsequently,each time two visible arcs are eliminated six edges are created (by |a(Pi)|)in the corresponding ternary tree in a natural way (see Figure 7 for anillustration). Our implicit rule for assigning the subtrees ensures that thefinal image is a locally symmetric ternary tree.Since final terms are reduced to patterns (a), (b), (c) and (d) in the processof executing the algorithm, it suffices to note that there is a bijection be-tween the patterns and certain parts of (growing) locally symmetric ternarytrees. Notice that (a) is unique since it occurs only when n is odd, at thefirst elimination. By considering the contents, especially the empty cases,

6

of Pi, τ(b), τ(c) and τ(d), we find that the resulting trees are distinct.We construct the inverse map of τ . For a given locally symmetric ternarytree, we read the internal vertices of subtrees from the bottom level to thetop level. This enables the construction of the corresponding final termoutwards by reversing the elimination algorithm as follows. For two identi-cal subtrees in the same level, apply a reverse of the multi-cases of pattern(a), (b) or (c), to insert the corresponding visible arcs. For two consecu-tive subtrees at two consecutive levels, apply the reverse of the single case.Lastly, an occurence of a single internal vertex, which is the root of the tree,corresponds the single visible arc of pattern (a). Thus, we have shown thatτ is a bijection.

Theorem 2.2 There is a bijection τ between final terms deduced by (1.2)with n arcs and locally symmetric ternary trees with n internal vertices.Moreover, the number of final terms with n arcs is

N3(x12x23 · · ·xn n+1) = s3n =

1n+1

(3n/2n/2

)

, if n is even,

2n+1

((3n−1)/2(n−1)/2

)

, if n is odd.

(2.1)

The bijection is illustrated in Figure 7. The visible arcs (about to be

1 2 3 4 5 6 7 8←→(a) +

1 2 3 4 5 6 7

←→(c) +2 3 4 5 6

←→(d) +4 5 6

←→(b)

Figure 7: The process of the bijection τ .

eliminated) are drawn with dash lines, and the pattern names are shownover the arrows. In the growing locally symmetric ternary subtrees, normal

7

vertices are drawn with solid dots, while the active vertices are drawn withopen circles.

Note that

• Deutsch, Feretic and Noy [3] considered symmetric ternary trees asfollows. For a ternary tree T , define its reflection T r by interchangingrecursively left and right children at every vertex. Then T is sym-metric if T = T r. They showed that the number of the symmetricternary trees with n internal vertices is enumerated by (2.1). Essen-tially this is the same as our locally symmetric trees by an obviousbijection, as indicated in Figure 8.

T1 T2 T2

←→

T2 T1T r2

Figure 8: The bijection between symmetric and locally symmetric ternarytrees.

• Theorem 2.2 may also be derived following repeated application of thereduction rule xijxjkxkh → xihxijxkh + xihxjkxkh, i < j < k < h, toa given final term. The linear representation is shown below.

−→ +i j k h i j k h i j k h

3 Reduction rule for locally symmetric k-ary

trees

In this section we extend rules (1.1) and (1.2), and give a bijection betweenthe final terms of (3.1) and locally symmetric k-ary trees. We begin with amonomial x = x12x23 · · ·xn n+1, and apply the following reduction rule forany positive integers i1 < i2 · · · < ik < ik+1.

xi1 i2 · · ·xik ik+1→ xi1 ik+1

xi1 i2 · · ·xik−1 ik+ xi1 ik+1

xi2 i3 · · ·xik ik+1, (3.1)

Denote the set of linear representations of all the resulting final termsby N k(x), and its cardinality by Nk(x) = |N k(x)|. A h-crossing of afinal term is defined as a set of h arcs (i1, j1), (i2, j2), · · · , (ih, jh) such thati1 < i2 < · · · < ih < j1 < j2 < · · · < jh. Note that a 1-crossing reduces toan arc.

8

· · · −→ · · · + · · ·i1 i2 ik ik+1 i1 i2 ik−1 ik ik+1 i1 i2 i3 ik ik+1

Figure 9: Linear representation of (3.1).

The application of (3.1) coupled with mathematical induction, gives thethe following properties for final terms with n arcs.

(1) The linear representation is a connected graph with n + 1 verticeslying on the x-axis, and n edges, such that no edge passes below the x-axis.

(2) Let n > 0. If n ≡ ℓ (mod k − 1) with 1 ≤ ℓ ≤ k − 2, there are ℓvisible arcs. If n ≡ 0 (mod k−1), there are k−1 visible arcs. Moreover, bythe elimination algorithm, the deletion of visible arcs at each level, alwaysreturns k − 1 new visible arcs in the reduced final (non-empty) term.

(3) We isolate the patterns which arise with the inductive action ofthe elimination algorithm namely (a), (bj) and (cj), where 1 ≤ j ≤ ℓ.The pattern (a) in Figure 10 applies to the first elimination when n ≡1 (mod k − 1), and P1 and P2 may be empty.

P1 P2

(a)

←→

v1

θ(a)

P1 P2 P2 · · · P2

Figure 10: Final terms in N k(x) and locally symmetric k-ary trees.

(4) The pattern (bj) with a j-crossing, followed immediately by ℓ − jconsecutive arcs, and the pattern (bℓ+1), are illustrated in Figure 11. Notethat pattern (bj) applies only to the first elimination when n ≡ ℓ (mod k−1), 2 ≤ ℓ ≤ k− 2. In the first figure Pi (1 ≤ i ≤ j− 1) may be empty whilePj is nonempty, in the second figure Pi (1 ≤ i ≤ ℓ) may be empty whilePℓ+1 is nonempty.

(5) The Multi cases are shown in Figure 12. Notice the relations withthe patterns in Figure 11. If n ≡ 0 (mod k − 1), then pattern (cj) may bechanged to (bj) by setting ℓ = k−1, where 1 ≤ j ≤ k−1. Similarly pattern(ck) corresponds to pattern (bℓ+1) as single cases.

The following property is a consequence of the uniqueness of the pat-terns and induction on the number of arcs.

Lemma 3.1 The final terms deduced from (3.1) are closed.

We are now prepared to state the desired bijection which can be estab-lished by means of the illustrations in Figure 10, Figure 11 and Figure 12,

9

· · ·P1 P2· · ·Pj−1 Pj

· · ·h h + 1

(bj)

←→v1

θ(bj)P1 P2 P2 · · ·P2

v2

P3P3 · · ·P3

vj−2

vj−1

PjPj · · ·Pj

vℓ

· · ·

P1 P2· · ·Pℓ−1 Pℓ Pℓ+1

· · ·h + 1

(bℓ+1)

←→v1

θ(bℓ+1)P1 P2 P2 · · ·P2

v2

P3P3 · · ·P3

vℓ−1

vℓ

Pℓ+1Pℓ+1· · ·Pℓ+1

Figure 11: Final terms in N k(x) and locally symmetric k-ary trees.

· · ·P1 P2· · ·Pj−1 Pj

· · ·h h + 1

(cj)

←→v1

θ(cj)P1 · · ·Pj · · ·

· · ·vk−1

P1 · · ·Pj · · ·

P1 P2· · ·Pk−2Pk−1 Pk

· · ·h + 1

(ck)

←→v1

θ(ck)P1 P2 P3 · · ·Pk

· · ·vk−1

P1 P2 P3 · · ·Pk

Figure 12: Final terms in N k(x) and locally symmetric k-ary trees.

in analogy with the proof of Theorem 2.2.

Theorem 3.2 There is a bijection θ between final terms deduced by (3.1)with n arcs and locally symmetric k-ary trees with n internal vertices, hence

Nk(x12x23 · · ·xn n+1) = skn =

ℓ+ 1

m

(

km+ ℓ

m− 1

)

(3.2)

for all n = m(k − 1) + ℓ with 0 ≤ ℓ ≤ k − 2.

Figure 13 gives an illustration of the bijection θ for k = 5.Remarks: If k = 2, the bijection θ reduces to the bijection given byGelfand, Graev and Postnikov [6], and s2n = Cn.

If k = 3, we obtain the bijection τ of Theorem 2.2.Lastly, we note that, besides (3.1), Theorem 3.2 can also be derived

from the following alternative rule.

xi1 i2 · · ·xik ik+1

→ xi1 ik+1xi1 i2 · · ·xik−2 ik−1

xik ik+1+ xi1 ik+1

xi2 i3 · · ·xik ik+1,

(3.3)

10

1 2 3 4 5 6 7 8 9 10 11 12

←→(b3)

+3 4 5 6 7 8 9 10 11

←→(c2)

+4 5 6 7 8

←→(c1)

Figure 13: The process of the bijection θ.

4 Reduction rule for symmetric even trees

We highlight the relationship between final terms and related combinatorialstructures studied by Deutsch, Feretic and Noy in [3], by giving a bijectionbetween final terms and symmetric even trees directly.

An even tree is a plane tree in which each vertex has even outdegree.Recall that a ternary tree is a plane tree in which each vertex has outdegree0 or 3. A noncrossing tree is a plane tree having as vertices a set of pointson the boundary of a circle and whose edges do not cross [4, 10]. Each of thethree combinatorial families admits a natural definition of symmetry. Takean even tree T for example, and define its reflection T r as the tree obtainedby recursively interchanging the left and right children at each vertex. Wesay that T is symmetric if T = T r. It is known (see [3]) that symmetriceven trees with 2n edges, symmetric ternary trees with n internal vertices,and symmetric noncrossing trees with n edges, are all enumerated by s3n.So there should be a bijection between final terms and symmetric eventrees.

The bijection is obtained by applying the elimination algorithm as pre-viously, but with a little modification. An illustration is given in Figure 14,where the P2 in φ(a) identifies v with the root of the corresponding subtree.Observe that (d), together with (e), constitute a refinement of pattern (d)in Figure 5. As before, the assignment of subtrees ensures that we ob-tain a symmetric even tree. Hence we have the following result, with thestraightforward proof omitted.

11

P1 P2

(a)

↔

v

φ(a)P1 P1

P2 P2P1 k k + 1

(b)

↔

v v′

φ(b)P1 P1

P1 P2 k k + 1

(c)

↔v v

′

φ(c)P1 P1P2 P2

P1 P2 P3 k k + 1

(d)

↔v v

′

P1 P1P2 P2

+P3 k k + 1

↔v v

′

φ(d)P1 P1P2 P2P3 P3

P1 P2 P3 P4 k k + 1

(e)

↔v v

′

P1 P1P2 P2

+P3 P4 k k + 1

↔v v

′

φ(e)P1 P1P2 P2P3 P3P4 P4

Figure 14: Final terms in N 3(x) and symmetric even trees.

Theorem 4.1 There is a bijection φ between final terms deduced from(1.2) with n arcs and symmetric even trees with 2n edges.

An example of the above bijection φ is given in Figure 15.It is known that the leftmost path directed from the root (leftmost path,

for short) plays an important role in the decomposition of plane trees (see[7, pp. 121-124] for details). From Theorem 4.1 we deduce at once thatthe number of arcs incident with the leftmost vertex in a final term equalsthe length of the leftmost path in the corresponding symmetric even tree,where the length of a path is the number of edges traversed. We see thatthe generating function of symmetric even trees with the leftmost path

of length k is xk(

xT 2(x2))k−1

S(x), where T (x) and S(x) are generatingfunctions of even trees and symmetric even trees respectively according tothe number of edge pairs which satisfy T (x) = 1 + xT 3(x) and S(x) =1 + xS(x)T (x2) = T (x2) + xT 2(x2) with [xn]S(x) = s3n, as stated in (2.1).Applying the Lagrange inversion formula to these equations, we obtain

T 2k−2(x) =

∞∑

n=0

2k − 2

n

(

3n+ 2k − 3

n− 1

)

xn. (4.1)

Now, making use of relations (2.1), (4.1) and the bijections of Deutsch,Feretic and Noy [3], we obtain a four-way equidistribution result.

12

1 2 3 4 5 6 7 8 9 10 11 12

←→(a)

+1 2 3 4 5 6 7 8 9 10 11

←→(e)

+2 3 4 5 6 7 8 9 10 11

←→(b)(c)

+6 7 8 9 10

←→(d)

Figure 15: The process of the bijection φ.

Theorem 4.2 The following classes of combinatorial objects are equinu-merous:

(i) Symmetric even trees with 2n edges and length of the leftmost pathk,

(ii) Symmetric ternary trees with n internal vertices and length of theleftmost path k,

(iii) Symmetric noncrossing trees (with representation in [11]) with n

edges and length of the leftmost path k,(iv) Final terms with n arcs k of which are incident with the leftmost

vertex.The common number of objects in each class is given by

s3n−1, if k = 1,

n−2k−1∑

i=0

4(k−1)n−2k−i+1

((3n−2k−3i−3)/2(n−2k−i−1)/2

)

s3i , if k ≥ 2,

where

s3n =

1n+1

(3n/2n/2

)

, if n is even,

2n+1

((3n−1)/2(n−1)/2

)

, if n is odd,

13

and binomial coefficients without integral indices are assumed to be 0.

5 Reduction rules for other Classes of Plane

Trees

The last section is devoted to a brief discussion of two further reductionrules. As in previous sections, we obtain bijections between the resultingfinal terms and two species of trees namely Motzkin trees and trees withloops. We state only the main results.

A Motzkin tree is a plane tree with n edges in which no vertex, the rootexcepted, has outdegree 1. It is known that that the number of such treesis the Motzkin number Mn ([5, 16], [16, Exerice 6.38]). Beginning with therule

xijxjkxkh → xihxijxjk + xikxjkxkh, for i < j < k < h, (5.1)

we follow the procedure of Section 3, with obvious adjustments, to classifythe properties of final terms and isolate arc partterns. Then we derive theassertion on the closure of final terms, leading to the following theorem.

Theorem 5.1 There is a bijection ψ between final terms deduced by (5.1)with n arcs and Motzkin trees with n edges. Hence, N(x12x23 · · ·xn n+1) =Mn.

We remark that for positive integers i < j < k < h, the following rule isequivalent to (5.1), in the sense that it also yields Theorem 5.1:

xijxjkxkh → xijxjhxkh + xijxjkxjh

orxijxjkxkh → xijxihxkh + xikxjkxkh,

Lastly we present a reduction rule which is related to plane trees withloops. These are plane trees in which a vertex may have at most one loop.For convenience, loops are drawn on the right sides of vertices. We assign aweight of 2 to each edge and 1 to each loop. The weight of a tree is the sumof the weights of all its edges and loops. By the bijection M3 in [5], the setof plane trees with loops and weight n is in one-to-one correspondence withthe set of Motzkin paths of length n with each horizontal step followed bya down-step unless the horizontal step ends the path. Such trees are areenumerated by Sequence A007477 in [14].

Now define the rule

xijxjkxkh → xihxijxjk + xijxjhxkh, for i < j < k < h. (5.2)

Then it can be shown, as in the previous cases, that the following as-sertion holds.

14

1 2 3 4 5 6 7 8 9 10 11

←→(a)

+1 2 3 4 5 6 7 8 9 10

←→(b) +

3 4 5 6 7 8 9 10

←→(b)

+

3 4 5 7 8 9 10

←→(b)(a)

+7 8 9

←→(b)

Figure 16: The process of the bijection ρ.

Theorem 5.2 There is a bijection ρ between final terms deduced by (5.2)with n arcs and the set of plane trees with loops of weight n. HenceN(x12x23 · · ·xn n+1) gives the Sequence A007477 in [14].

Figure 16 illustrates the realization of the asserted bijection.Acknowledgments. The authors would like to thank William Y.C.

Chen, Alex Postnikov, and Andreas Dress for their helpful discussions andsuggestions. Also the authors would like to thank Rebecca Smith for read-ing a previous version of the present paper.

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