On Thu, Jul 16, 2009 at 2:18 AM, Ross Levine<ross.levine@uky.edu> wrote:
On Thu, Jul 16, 2009 at 12:26 AM, Gottlob Frege <gottlobfrege@gmail.com>wrote:
What are the requirements of 'tree-ness'? Containers like std::map/list/deque/vector/... all have well defined requirements.
This is one of those questions that seems trivial but is very important to actually lay out.
Glad you agree. Many people I work with just like to run ahead and (blindly, in my opinion) type code as fast as they can! <cut/paste>
Someone check my work.
OK! My pure-math proof/theorem/formalism hat popped onto my head, but I'll try not to get too picky.
Here's what I can come up with:
1. A tree is a node-based container. If nonempty, every node has exactly one parent, except for one node. This node is called the _root node_.
node, parent, container, etc undefined. I can live with that. Can't define everything!
2. Every node has a finite, non-negative number of children. A node's children is the same as the set of nodes which have that node as a parent.
In fact, this is the definition of 'child'.
Corollary: There is exactly one simple path between any two different nodes (a simple path is a path that has no repeated vertices).
vertex undefined but I think you meant 'node' :-) simple path undefined this is actually worth defining, somewhat (in my head at least), because, in addition to these requirements, there is nothing stopping a tree from having 'extra' edges (ie linking children together) but obviously these don't form a part of the 'tree-ness' so shouldn't/can't be used as part of a path. (that's just my head trying to imagine how the corrollary could be false, and closing the loop-holes)
Corollary: A node's parent lies on the path between it and the root node.
Definitions: 1. If any node in tree T have at most K children, then T is a _K-ary tree_.
'have at most' at the moment, or 'can have at most' at any time? Maybe it depends on context? (ie T is, at the moment, a K-ary tree...? Or would it better to say, in those case, "T is, at the moment, *like* a K-ary tree, or can be considered a K-ary tree for the purposes of blah blah blah...")
2. A K-ary tree is _full_ if every node has either 0 or K children. 3. The _distance_ between two nodes, A and B, is the number of edges in the path that connects A and B. 4. The _height_ of tree T is equal to the maximum distance from the root node to any other node. 5. The _degree_ of a node is the number of children, plus the number of parents. Since every node (except the root) has exactly one parent, the degree of a non-root node is the number of children plus 1.
what's 'degree' typically used for? is it more for graphs? Seems useless as a concept of its own, separate from 'number of children', at least for trees. (just wondering)
6. A node is a _leaf_ node if it has no children. 7. An _ordered tree_ is a tree in which the position of the nodes, and the order of a node's children, is significant.
That's all I can think of right now. Someone check my work.
excellent stuff. I can't think of any other requirements - anything I could consider adding just makes it a specialization of tree. I guess it might be useful to mention ('corollary') that a tree is thus a type of graph, more specifically a type of DAG. I'm trying to imagine if there are interesting cases between 'tree' and 'DAG' that are worth thinking about... Tony