CompleteKaryTree[n, k] returns a complete k-ary tree on n vertices.
HighlightedEdgeColors is an option to Highlight that determines which colors are used for the highlighted edges.
HighlightedEdgeStyle is an option to Highlight that determines how the highlighted edges are drawn.
HighlightedVertexColors is an option to Highlight that determines which colors are used for the highlighted vertices.
HighlightedVertexStyle is an option to Highlight that determines how the highlighted vertices are drawn.
LabeledTreeToCode[g] reduces the tree g to its Prüfer code.
LoopPosition is an option to ShowGraph whose values tell ShowGraph where to position a loop around a vertex. This option can take on values UpperLeft, UpperRight, LowerLeft, ...
NormalizeVertices[v] gives a list of vertices with a similar embedding as v but with the coordinates of all points scaled to be between 0 and 1.
RandomTree[n] constructs a random labeled tree on n vertices.
RankGraph[g, l] partitions the vertices into classes based on the shortest geodesic distance to a member of list l.