I am writing work on trees in graph theory and a bit confused in concepts. I need to write what kinds of trees exist or the classification of trees. I have found that there exists: oriented / non-oriented tree, spanning, forest, binary, n-dimensional, ordered tree. But it is more and more similar to the properties of trees than to species (except for binary and n-dimensional).
What of this (if I correctly found and missed nothing) will really be a separate type of tree? Also, there is for example a special case of a binary tree - the search tree. Will such a tree be related to graph theory or is it more data structure in programming? The same question with the parse tree. And the last question: does tree walking refer to tree operations as graphs or is the algorithm for the data structure a tree?