[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.
Franklin T. Adams-Watters
-----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:
http://www.cetteadressecomportecinquantesignes.com/HLT.htm
Best,
É.
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