All Questions
Tagged with enumerative-combinatorics trees
14 questions
18
votes
2
answers
1k
views
A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
- 1,190
6
votes
0
answers
276
views
Power law correction factor in tree enumeration via naïve division
It is a theorem of Otter, building on fundamental work of Pólya, that the number of unlabeled trees on $n$ vertices is $\approx C \alpha^{n} n^{-5/2}$, where $C = 0.534\ldots$ and $\alpha = 2.955\...
- 24.2k
14
votes
1
answer
647
views
Bijective proof of recurrence for rooted unlabeled trees
Would've been a better question for Christmas than Thanksgiving, but alas...
Let $t_n$ denote the number of rooted, unlabeled trees on $n$ vertices (OEIS A000081). These are the isomorphism classes of ...
- 24.2k
1
vote
0
answers
85
views
Varieties of trees with logarithmic degree function
I am interested in varieties of trees with a logarithmic degree function. I am currently looking at Bergeron, Flajolet, and Salvy's work "Varieties of increasing trees." They discuss exactly ...
- 81
7
votes
1
answer
2k
views
How does the number of trees on $n$ vertices *up to isomorphism* grow as $n \to \infty$?
It is well known that the number of labelled trees on $n$ vertices is equal to $n^{n-2}$.
We do not expect any such exact formula for the number of isomorphism types of trees on $n$ vertices. But ...
- 7,857
1
vote
1
answer
248
views
mapping integers to k-ary trees
Is there an algorithmic way to map the natural numbers to unique k-ary trees?
I am familiar with the work of Tychonievich who created a mapping from integers to binary trees. https://www.cs.virginia....
- 113
7
votes
1
answer
571
views
Are there Prüfer sequences for rooted forests?
One well-known, extremely slick proof of Cayley's tree enumeration theorem is the use of Prüfer sequences. Cayley also proved a version for forests, namely that the number of forests with $n$ ...
- 383
7
votes
1
answer
705
views
Is there a natural relationship between OEIS A127670 and Cayley's tree formula?
I apologize in advance that this question must sound highly amateurish, but I am wondering if there is any connection between the formula https://oeis.org/A127670 , which counts the number of fixed $n$...
- 4,049
5
votes
2
answers
351
views
Asymptotics of unrooted labeled forests
It is well known that the number of unrooted labeled trees on vertex set
$[n]={1,2,...,n}$ is $n^{n-2}$. Let $U(z)$ be the exponential generating function of the sequence of these numbers. Then $F(z)=\...
- 371
12
votes
0
answers
330
views
The number of labeled pairs of edge disjoint trees and related questions
I wonder what is known on the following:
1) What is the number $T_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ with $n$ labelled vertices?
2) (harder, it seems) What ...
- 24.7k
8
votes
1
answer
344
views
Bijective proof of formula for rooted binary forests
For $n\ge 1$, let $f(n)$ be the number of rooted complete (unordered) binary trees with $n$ leaves labeled from $1$ to $n$ ("complete binary" means that every vertex has either $0$ or $2$ children and ...
- 82.6k
7
votes
1
answer
455
views
More asymptotics for trees
This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
- 1,305
14
votes
1
answer
696
views
Are the asymptotics of A003238 known?
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...
- 1,305
13
votes
5
answers
1k
views
Is the following invariant of rooted trees a complete invariant?
Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex ...
- 1,325