# On a Class of Extremal Trees for Various Indices — Online Ressources

This page contains online ressources for the paper “On a Class of Extremal Trees for Various Indices” by C. Heuberger and S. Wagner.

## Asymptote Package

In festoontree.asy, we provide an Asymptote package for drawing F-trees.

Currently (as of asymptote 1.43), a patched copy of drawtree.asy is required.

Usage: Copy the two files into one directory. A sample usage would be (download)

```import festoontree;
FestoonTree(3,69);
```

This creates the F-tree for d=3 and n=69 in explicit form.

To get a decomposed form, use (download)
```import festoontree;
FestoonTree(3,69,true);
```

Running `asy` on these files yields .eps files containing the graphs.

## Sample Files

For up to 100 vertices and d at most 5, we have compiled pdf files containing the F-trees.

d Decomposed Form Explicit Form
2 305 KB 162 KB
3 311 KB 151 KB
4 316 KB 153 KB
5 320 KB 149 KB

## Mathematica® Files

The file FestoonTree.m contains Mathematica® code for creating F-trees and computing their Merrifield-Simmons and Hosoya indices as well as their energy.

An examples file is also available.

```<<FestoonTree.m
```

```FestoonSequence[3,139]
FestoonTreeSymbolic[3,69]
FestoonTreeSemiSymbolic[3,69]
FestoonTree[3,69]
FestoonTree[3,69]//TreeForm
```

Computes the F-sequence as well as the F-trees. `FestoonSequence[d,N]` computes the d-ary F-sequence of N. `FestoonTreeSymbolic[d,n]` gives a symbolic decomposition of the d ary F-tree of order n, in `FestoonTreeSemiSymbolic` the Festoon is already replaced by Tree constructs, and in `FestoonTree`, the complete d-ary trees are also translated into Tree constructs.

`Tree[b1,...,bn]` denotes a rooted tree with branches b1, ..., bn.

```sigma[FestoonTree[3,69]]
z[FestoonTree[3,69]]
M[FestoonTree[3,69],x]
Energy[FestoonTree[3,69]]
N[Energy[FestoonTree[3,69]]]
```

Computing the Merrifield-Simmons index `sigma`, the Hosoya index `z`, the matching polynomial `M[T,x]` and the energy of a F-tree.

```sigma[FestoonTreeSemiSymbolic[3,69]]
z[FestoonTreeSemiSymbolic[3,69]]
M[FestoonTreeSemiSymbolic[3,69],x]
Energy[FestoonTreeSemiSymbolic[3,69]]
N[Energy[FestoonTreeSemiSymbolic[3,69]]]
```

Working with `FestoonTreeSemiSymbolic` (or `FestoonTreeSymbolic`) increases the performance of the index calculations, as intermediate results for complete trees are cached.

C. Heuberger, S. Wagner.
Last Modification: 2008-08-27