[seqfan] Re: Hetero-labeled trees

franktaw at netscape.net franktaw at netscape.net
Tue Dec 13 15:37:16 CET 2011

```It is IMO better to describe these as "weights" rather than "labels".
So what you are defining are "hetero-weighted trees", the requirement
is that each node has a distinct positive integer weight, and the
weight of any non-leaf node is the sum of the weights of its children.

As far as I know, no one has studied these, but I don't know everything.

Try counting how many such trees there are with a given weight at the
root, and see if that sequence is in the database. (One could break
this down further, asking how many such trees there are with leaf
weights a particular partition into distinct parts (see A118457). There
will always be at least one: the single node tree for a partition with
only one part, and a height 2 tree for any other partition into
distinct parts.)

Other than that, I have to wonder whether every term in your sequence
is a triangular number. Whether it is or not, this suggests another
sequence: the maximum number of consecutive values 1 to a(n) in a tree
whose leaves have weights 1 to n.

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>

Hello SeqFans,
An hetero-labeled tree is a tree showing no two identical labels:

10
+--+--+     <------ top
|     |
6     |     <-.
+--+--+  |        \
|     |  |         \
5     |  |     <------ intermediate results
+--+--+  |  |
|     |  |  |
2     3  1  4     <------ ground

We are interested in HLT, like the one above, which are provided
with an additive rule: integers on the ground are added, at some
point (as are the intermediate ones), but the integer on top is
unique.

Question:

What is the smallest integer on top for HLTs showing all integers
from 1 to n (plus others, if needed - but integers from 1 to n
must appear in the tree)?

The "smallest top integers" sequence seems to start like this
(not in the OEIS):

S = 1, 3, 3, 6, 10, 10, 15, 15,...

I'm definitely lost for bigger values of n... Is there an algorithm?
A formula?
Sorry if this is old hat -- trees must have been searched a lot,
I guess.

More examples here:
Best,
É.

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