# [seqfan] Hetero-labeled trees

Eric Angelini Eric.Angelini at kntv.be
Mon Dec 12 18:13:11 CET 2011

```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: