A new bijection on rooted forests
Discrete Mathematics North-Holland
179
111 (1993) 1799188
A new bijection
on rooted forests
Pkter L. Erdiis* Institute of Operations Reseurch, Unirersity of Bonn, Bonn, Germany Mathematical Institute qf Hungarian Academy qf Science, Budapest, Hungar? Recieved
22 July 1991
Abstract Erdos,
P. L., A new bijection
on rooted
forests, Discrete
Mathematics
111 (1993) 179-188.
This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees. A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). This method yields new easy proofs for some well-known theorems which use only elementary calculations with the sums of Stirling numbers.
l_ Preliminaries The theory of combinatorial enumerations determines the cardinality of collections of certain objects. But, usually, it is considered to be a more useful result if one can give a concrete one-to-one correspondence between the objects and another, wellknown, set. The goal of this paper is twofold. Firstly, we determine a bijection on a certain subclass of rooted forests (called the semilabelled forest). Secondly, we would like to show a good candidate to be a basic structure of further forest enumeration problems. This means that on the basis of the semilabelled forests, the proof of some previously known enumerative problem can be transformed into a very simple calculation with sums of Stirling numbers of second kind. This paper can be considered as an extension of the paper by Szekely and Erdiis [2] on the enumeration of trees. That paper presented a general bijection between the set of semilabelled trees and some sets of partitions. (Throughout the present paper, this one-to-one correspondence will be referred to as the ST-bijection.) For the theory of enumeration of trees, see [S, 71. In this section we describe all the necessary definitions and results from [2]. The ST-bijection is based on the squashed order of subsets of an ordered set, which was introduced into the finite set theory independently by Kruskal [6] and Katona [4]. Correspondence to: P.L. Erdds, Hortensiastraat 3, 1338 ZP Almere, Netherlands * Research supported by Alexander Y. Humboldt-Stiftung. 0012-365X/93/$06.00
‘c
1993-Elsevier
Science Publishers
B.V. All rights reserved
180
P.L. Erdds
Definition 1.1. Let X be an ordered X is defined as follows: A the set of leaves i
for v~(x,, separated
from r by v
x2,...,xk};
otherwise.
It may happen that some sets S(u) occur with multiplicity, namely, if some branching points have out-degree one. Therefore, we use
of an (n-m
+ l)-elements
k-m),
due to Corollary 1.4(a). (b) If the forest F has no trivial tree, then the multiset M(F) is a proper set whose elements are branching points of the semilabelled tree T(F). The number of such sets M(F) is the binomial coefficient in the statement. (c) The proof of this claim is an elementary calculation with Stirling numbers of second kind. Here we give it only as a prototype: f(n,k,.)=~~f(n,k,m)=~~ In the second tion 1.2.61):
step we applied
S(a, b)= 2
(iI:)S(a-m,
(EIi)S(n-m,n-k)=S(n,n-k+l). the following
b-l).
well-known
0
equation
(e.g. 5, Sec-
(2)
m=l
3. Further easy consequences Here, we list some further results. The proofs are so easy that they are (mainly) omitted. Each of them involves elementary calculations with sums of Stirling numbers, applying sometimes the equation (2) or the equation i
S(a, k)b(b-l)...(b-k+
l)=b”.
k=l
We must deduce the special properties of the SF-bijection only sometimes. Just that simplicity can motivate (according to our judgement) the new proofs of these (mainly)
A new bijection on rooted forests
185
well-known theorems. The given references are (generally) from the excellent book of Moon [7]. We remark that these results can be proved in several ways (for example, by applying the Priifer method), but the original proofs were based (in general) on the generating-function
method.
Corollary 3.1. The number of labelled
rootedforests
S(n -m,
with n points, k leaves and m trees is
n - k).
Proof. We select k labels out of n in (z) ways for the leaves, construct (iT:)S(n-m, n-k) semilabelled forests for each label set by Corollary 2.3(a), and distribute the remaining n-k labels for the nonleaf vertices in (n-k)! ways. 0 Corollary 3.2. The number of labelled lf(n,k,.)=
rooted forests
with n points
and k leaves
is
lf(n.k,m)=gS(n,n-k+l).
t IfI=1
Corollary 3.3 (Riordan m trees
[lo]).
The number
of labelled
rooted forests
with n points
and
is n-1
d(n, . , m) = ~lf(n,k,m)=(~~~)nn-m=(~)mnn-m-l. Corollary 3.4 (Gtibel [3]). The number of labelled forests where the jixed vl,. . . , v, points belong to distinct trees is
with n points
and m trees
mnn-m-l
Proof. When we build up rooted labelled forests on n points having m trees, we can select the labels of the roots in (i) ways. The number of labelled forests having a concrete root system is independent of the labels of the number of labelled forests containing a concrete root system of labelled rooted forests. We can consider the vertices vl, system and, by Corollary 3.3, the number of labelled forests system is 1
n-l
(3
( m-l
-
n n-m=,,n-m-l~
roots. Consequently, the is l/(i) times the number . . . . v, as a concrete root which contain this root
q
1
Corollary 3.5 (Riordan lf(n ,.,. )= i k=l
[lo]).
The number
lf(n,k,.)=(n+l)“-‘.
of rooted
labelled
forests
with n points
is
186
P.L. Erdh
Now we study forests with restrictions. (We use this notion due to Riordan [ll].) These forests consist of special trees. These restrictions can be given with parameters which are not related to the labelling process. For example, such parameters are the height of the trees, or the out-degrees
of the points.
Under the SF-bijection, some of them are in correspondence to special trees (and through the ST-bijection, to special partitions). For example, if the forest F has binary trees, then T(F) is a binary Corollary 3.6 (Carter forests
tree.
et al. Cl]). Zf b(k, m) denotes the number of binary semilabelled
with k leaves and m trees, then
Proof. As stated above, it is trivial to prove that if F is a binary semilabelled forest then the semilabelled tree T(F) made by the SF-bijection is a binary tree. We know that I( T(F)) = k - m + 1. Consequently, v( T(F)) = 2k - 2m + 1 and v(F) = 2k - m. Applying Corollary 1.4(b) and Theorem 2.1, we obtain the desired formula. 0 The following problems were suggested by Riordan. He studied the labelled (and nonlabelled) rooted trees with restricted height parameter in [9]. The height of a rooted tree is the edge number of the longest path from the root. (A similar notion for unrooted tree is the diameter.) Riordan extended the results of the quoted paper for forests having height-restricted trees ([ll]). The highest tree with a given number of vertices is the path rooted at one of its endpoints. The least height of a tree is 0 if the trees has one point, and is 1 otherwise. The trees with height 1 are the stars rooted at the central point. Corollary 3.7 (Riordan [ll]). (a) Th e number of semilabelled forests with n points, k leaves and m trees which are paths rooted at their endpoints (one-degree points) is
p(n,km)=
(lri 1.
(b) The number of the labelled forests
lp(n,k,m)=$
:I: (
of the same kind is
. >
Proof. It is easy to see that T(F) is a path rooted at one endpoint. The number of semilabelled trees of this kind is one, independent of the parameters 1(F) or m(F). That gives (a). Since in these forests 1(F) = m(F), (b) can be proved by the same argument as Corollary 3.1. 0
A new bijection on rooted forests
Corollary 3.8. The number of semilabelled forests which are stars rooted at the central points is S(n -m,
187
with n points, k leaves and m trees
n - k).
This kind of forest will be referred
to in this paper as star forest.
Proof of Corollary 3.8. Let F be a star forest. It is easy to see that k>m,
kan-m.
If T(F) has at least 2 vertices T(F) is v(T(F))-l(T(F))-
(i.e. m
179
111 (1993) 1799188
A new bijection
on rooted forests
Pkter L. Erdiis* Institute of Operations Reseurch, Unirersity of Bonn, Bonn, Germany Mathematical Institute qf Hungarian Academy qf Science, Budapest, Hungar? Recieved
22 July 1991
Abstract Erdos,
P. L., A new bijection
on rooted
forests, Discrete
Mathematics
111 (1993) 179-188.
This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees. A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). This method yields new easy proofs for some well-known theorems which use only elementary calculations with the sums of Stirling numbers.
l_ Preliminaries The theory of combinatorial enumerations determines the cardinality of collections of certain objects. But, usually, it is considered to be a more useful result if one can give a concrete one-to-one correspondence between the objects and another, wellknown, set. The goal of this paper is twofold. Firstly, we determine a bijection on a certain subclass of rooted forests (called the semilabelled forest). Secondly, we would like to show a good candidate to be a basic structure of further forest enumeration problems. This means that on the basis of the semilabelled forests, the proof of some previously known enumerative problem can be transformed into a very simple calculation with sums of Stirling numbers of second kind. This paper can be considered as an extension of the paper by Szekely and Erdiis [2] on the enumeration of trees. That paper presented a general bijection between the set of semilabelled trees and some sets of partitions. (Throughout the present paper, this one-to-one correspondence will be referred to as the ST-bijection.) For the theory of enumeration of trees, see [S, 71. In this section we describe all the necessary definitions and results from [2]. The ST-bijection is based on the squashed order of subsets of an ordered set, which was introduced into the finite set theory independently by Kruskal [6] and Katona [4]. Correspondence to: P.L. Erdds, Hortensiastraat 3, 1338 ZP Almere, Netherlands * Research supported by Alexander Y. Humboldt-Stiftung. 0012-365X/93/$06.00
‘c
1993-Elsevier
Science Publishers
B.V. All rights reserved
180
P.L. Erdds
Definition 1.1. Let X be an ordered X is defined as follows: A the set of leaves i
for v~(x,, separated
from r by v
x2,...,xk};
otherwise.
It may happen that some sets S(u) occur with multiplicity, namely, if some branching points have out-degree one. Therefore, we use
of an (n-m
+ l)-elements
k-m),
due to Corollary 1.4(a). (b) If the forest F has no trivial tree, then the multiset M(F) is a proper set whose elements are branching points of the semilabelled tree T(F). The number of such sets M(F) is the binomial coefficient in the statement. (c) The proof of this claim is an elementary calculation with Stirling numbers of second kind. Here we give it only as a prototype: f(n,k,.)=~~f(n,k,m)=~~ In the second tion 1.2.61):
step we applied
S(a, b)= 2
(iI:)S(a-m,
(EIi)S(n-m,n-k)=S(n,n-k+l). the following
b-l).
well-known
0
equation
(e.g. 5, Sec-
(2)
m=l
3. Further easy consequences Here, we list some further results. The proofs are so easy that they are (mainly) omitted. Each of them involves elementary calculations with sums of Stirling numbers, applying sometimes the equation (2) or the equation i
S(a, k)b(b-l)...(b-k+
l)=b”.
k=l
We must deduce the special properties of the SF-bijection only sometimes. Just that simplicity can motivate (according to our judgement) the new proofs of these (mainly)
A new bijection on rooted forests
185
well-known theorems. The given references are (generally) from the excellent book of Moon [7]. We remark that these results can be proved in several ways (for example, by applying the Priifer method), but the original proofs were based (in general) on the generating-function
method.
Corollary 3.1. The number of labelled
rootedforests
S(n -m,
with n points, k leaves and m trees is
n - k).
Proof. We select k labels out of n in (z) ways for the leaves, construct (iT:)S(n-m, n-k) semilabelled forests for each label set by Corollary 2.3(a), and distribute the remaining n-k labels for the nonleaf vertices in (n-k)! ways. 0 Corollary 3.2. The number of labelled lf(n,k,.)=
rooted forests
with n points
and k leaves
is
lf(n.k,m)=gS(n,n-k+l).
t IfI=1
Corollary 3.3 (Riordan m trees
[lo]).
The number
of labelled
rooted forests
with n points
and
is n-1
d(n, . , m) = ~lf(n,k,m)=(~~~)nn-m=(~)mnn-m-l. Corollary 3.4 (Gtibel [3]). The number of labelled forests where the jixed vl,. . . , v, points belong to distinct trees is
with n points
and m trees
mnn-m-l
Proof. When we build up rooted labelled forests on n points having m trees, we can select the labels of the roots in (i) ways. The number of labelled forests having a concrete root system is independent of the labels of the number of labelled forests containing a concrete root system of labelled rooted forests. We can consider the vertices vl, system and, by Corollary 3.3, the number of labelled forests system is 1
n-l
(3
( m-l
-
n n-m=,,n-m-l~
roots. Consequently, the is l/(i) times the number . . . . v, as a concrete root which contain this root
q
1
Corollary 3.5 (Riordan lf(n ,.,. )= i k=l
[lo]).
The number
lf(n,k,.)=(n+l)“-‘.
of rooted
labelled
forests
with n points
is
186
P.L. Erdh
Now we study forests with restrictions. (We use this notion due to Riordan [ll].) These forests consist of special trees. These restrictions can be given with parameters which are not related to the labelling process. For example, such parameters are the height of the trees, or the out-degrees
of the points.
Under the SF-bijection, some of them are in correspondence to special trees (and through the ST-bijection, to special partitions). For example, if the forest F has binary trees, then T(F) is a binary Corollary 3.6 (Carter forests
tree.
et al. Cl]). Zf b(k, m) denotes the number of binary semilabelled
with k leaves and m trees, then
Proof. As stated above, it is trivial to prove that if F is a binary semilabelled forest then the semilabelled tree T(F) made by the SF-bijection is a binary tree. We know that I( T(F)) = k - m + 1. Consequently, v( T(F)) = 2k - 2m + 1 and v(F) = 2k - m. Applying Corollary 1.4(b) and Theorem 2.1, we obtain the desired formula. 0 The following problems were suggested by Riordan. He studied the labelled (and nonlabelled) rooted trees with restricted height parameter in [9]. The height of a rooted tree is the edge number of the longest path from the root. (A similar notion for unrooted tree is the diameter.) Riordan extended the results of the quoted paper for forests having height-restricted trees ([ll]). The highest tree with a given number of vertices is the path rooted at one of its endpoints. The least height of a tree is 0 if the trees has one point, and is 1 otherwise. The trees with height 1 are the stars rooted at the central point. Corollary 3.7 (Riordan [ll]). (a) Th e number of semilabelled forests with n points, k leaves and m trees which are paths rooted at their endpoints (one-degree points) is
p(n,km)=
(lri 1.
(b) The number of the labelled forests
lp(n,k,m)=$
:I: (
of the same kind is
. >
Proof. It is easy to see that T(F) is a path rooted at one endpoint. The number of semilabelled trees of this kind is one, independent of the parameters 1(F) or m(F). That gives (a). Since in these forests 1(F) = m(F), (b) can be proved by the same argument as Corollary 3.1. 0
A new bijection on rooted forests
Corollary 3.8. The number of semilabelled forests which are stars rooted at the central points is S(n -m,
187
with n points, k leaves and m trees
n - k).
This kind of forest will be referred
to in this paper as star forest.
Proof of Corollary 3.8. Let F be a star forest. It is easy to see that k>m,
kan-m.
If T(F) has at least 2 vertices T(F) is v(T(F))-l(T(F))-
(i.e. m
