seed = integer or none

Seed for the random number generator. When an integer is specified, this is equivalent to calling randomize(seed).

weights = range or procedure

If the option weights=m..n is specified, where $m\le n$ are integers, the tree is a weighted graph with edge weights chosen from [m,n] uniformly at random.  The weight matrix W in the graph has datatype=integer, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.

If the option weights=x..y where $x\le y$ are decimals is specified, the tree is a weighted graph with numerical edge weights chosen from [x,y] uniformly at random.  The weight matrix W in the graph has datatype=float[8], that is, double precision floats (16 decimal digits), and if the edge from vertex i to j is not in the graph then W[i,j] = 0.0.

If the option weights=f where f is a function (a Maple procedure) that returns a number (integer, rational, or decimal number), then f is used to generate the edge weights.  The weight matrix W in the tree has datatype=anything, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.

The RandomTree(n)  command creates a random tree on n vertices. This is an undirected connected graph with n-1 edges. If the first input n is a positive integer, the vertices are labeled 1,2,...,n. Alternatively you may specify the vertex labels in a list.

Starting with the empty undirected graph T on n vertices, edges are chosen uniformly at random and inserted into T if they do do not create a cycle.  This is repeated until T has n-1 edges.

The option $\mathrm{degree}<d$ or $\mathrm{degree}\le d$ limits the maximum degree of every vertex in the tree.

The random number generator used can be seeded using the seed option or the randomize function.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\&colon;$

$\mathrm{with}\left(\mathrm{RandomGraphs}\right)\&colon;$

$T≔\mathrm{RandomTree}\left(10\right)$

$T≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 10\; vertices\; and\; 9\; edge(s)}$

$T≔\mathrm{RandomTree}\left(10\&comma;\mathrm{weights}=1..9\right)$

$T≔\mathrm{Graph\; 2:\; an\; undirected\; weighted\; graph\; with\; 10\; vertices\; and\; 9\; edge(s)}$

$\mathrm{IsTree}\left(T\right)$

$\mathrm{true}$

$\mathrm{WeightMatrix}\left(T\right)$

$\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 8& 0& 0\\ 0& 0& 0& 6& 0& 0& 5& 0& 7& 0\\ 0& 0& 0& 0& 0& 0& 0& 4& 0& 6\\ 0& 6& 0& 0& 0& 0& 0& 6& 0& 0\\ 0& 0& 0& 0& 0& 2& 0& 0& 0& 8\\ 0& 0& 0& 0& 2& 0& 0& 0& 0& 0\\ 0& 5& 0& 0& 0& 0& 0& 0& 0& 0\\ 8& 0& 4& 6& 0& 0& 0& 0& 0& 0\\ 0& 7& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 6& 0& 8& 0& 0& 0& 0& 0\end{array}\right]$

$T≔\mathrm{RandomTree}\left(100\right)$

$T≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 100\; vertices\; and\; 99\; edge(s)}$

$\mathrm{MaximumDegree}\left(T\right)$

$6$

$T≔\mathrm{RandomTree}\left(100\&comma;\mathrm{degree}<4\right)$

$T≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 100\; vertices\; and\; 99\; edge(s)}$

$\mathrm{MaximumDegree}\left(T\right)$

$3$