PLEASE ANSWER IN PYTHON NO CHAT GPT

Random Trees

The nx$.$random$\_$tree$($n$)$ command can be used to generate a random tree on the n vertices represented in python by range$($n$)($in order$).$

Such a tree can be converted into a sequence of n$2$

numbers, its Pr$$fer code by the python function pruefer$\_$code, defined as follows:

$**$given code$**$

def pruefer$\_$node$($tree$)$:

for x in tree:

if tree.degree$($x$)==1$:

for y in tree$[$x$]$:

tree.remove$\_$node$($x$)$

return y

def pruefer$\_$code$($T$)$:

tree $=$ T$.$copy$()$

return $[$pruefer$\_$node$($tree$)$ fo

code $=$ pruefer$\_$code$($T$)$

code

$**$

Conversely, the following function tree$\_$pruefer constructs a tree from a Pr$$fer code.

$**$Given code$**$

def tree$\_$pruefer$($code$)$:

# initialize graph and defects

n $=$ len$($code$)+2$

tree $=$ nx$.$empty$\_$graph$($n$)$

degrees $=[1$ for x in tree$]$

for y in code:

degrees$[$y$]+=1$

# add edges

for y in code:

for x in tree:

if degrees$[$x$]==1$:

tree.add$\_$edge$($x$,$ y$)$

for z in $($x$,$ y$)$:

degrees$[$z$]-=1$

break

# final edge

e $=[$x for x in tree if degrees$[$x$]==1]$

tree.add$\_$edge$(*$e$)$

return tree

T$2=$ tree$\_$pruefer$($code$)$

$**$

Question

Use, if needed, the above functions to experiment with random trees and$/$or sequences and answer the following:

$1.$Which labelled tree on n

nodes $\{0,1,...,$n$1\}$

corresponds to the Pr$$fer code $[0,1,2,...,$n$3]?$

$($Note: you need to describe and justify mathematically an answer for all n$.)$

$2.$More generally, what can one say about a tree n

whose Pr$$fer code consists of n$2$

distinct entries? $($Note: you need to describe and justify mathematically an answer for all n$.)$

$3.$ Which labelled tree on n

nodes $\{0,1,...,$n$1\}$

corresponds to a Pr$$fer code which has n$2$

identical entries?

$($Note: you need to describe and justify mathematically an answer for all n$.)$