# [seqfan] A002955: rooted trimmed trees

Joerg Arndt arndt at jjj.de
Mon Mar 2 17:55:42 CET 2015

```I observe that this sequence (seems to) count
"Unordered rooted trees without "x x" in the level
sequence for the pre-order walk."

Here are those (8) trees for n = 6:

1:  [ 0 1 2 3 4 5 ]
O--o--o--o--o--o

2:  [ 0 1 2 3 4 3 ]
O--o--o--o--o
.--o

3:  [ 0 1 2 3 4 2 ]
O--o--o--o--o
.--o

4:  [ 0 1 2 3 4 1 ]
O--o--o--o--o
.--o

5:  [ 0 1 2 3 2 3 ]
O--o--o--o
.--o--o

6:  [ 0 1 2 3 2 1 ]
O--o--o--o
.--o
.--o

7:  [ 0 1 2 3 1 2 ]
O--o--o--o
.--o--o

8:  [ 0 1 2 1 2 1 ]
O--o--o
.--o--o
.--o

Now I realize that I do _not_ understand the comments.
My understanding of "limb" is
"the way (subtree) from a leaf towards the root until
the first branch point (which can be the root) is reached."
Example: The tree
[ 0 1 2 3 4 2 1 2 1 ]
O--o--o--o--o
.--o
.--o--o
.--o
has limbs of lengths 4, 1, 2, and 1.

Now from what I read in A002955 it should count
trees without limbs of length 2 (? or length >= 2 ?).
The trees I show above are(?) different, they avoid subtrees
*--o
.--o

Can somebody clarify?

Best regards,   jj

P.S.: should "unordered" be added to the name?
P.P.S.: should "free" be added to the names of
sequences that are about free trees (like A000055)?
```