In the algorithm above, all cases are chained in order, except in delete case 3 where it can recurse to case 1 back to the parent node : this is the only case where an iterative implementation will effectively loop.
32.
In the second phase, the multiple unconnected trees are turned into a single tree by merging the roots of the initial trees, but this time starting from the right and adding new parent nodes as needed ( red nodes ).
33.
Nodes are placed closer to a baseline than their parent node, by a factor equal to the number of child nodes of that parent node ( or by some sort of weighting ), and scaled according to how close they are.
34.
Nodes are placed closer to a baseline than their parent node, by a factor equal to the number of child nodes of that parent node ( or by some sort of weighting ), and scaled according to how close they are.
35.
If an internal node has 2d keys, then adding a key to that node can be accomplished by splitting the hypothetical 2d + 1 key node into two d key nodes and moving the key that would have been in the middle to the parent node.
36.
Consequently a deterministic top-down tree automaton will only be able to test for tree properties that are true in all branches, because the choice of the state to write into each child branch is determined at the parent node, without knowing the child branches contents.
37.
While it may seem as though the operators have to be invertible for the reverse search, it is only necessary to be able to find, given any node n, the set of parent nodes of n such that there exists some valid operator from each of the parent nodes to n.
38.
While it may seem as though the operators have to be invertible for the reverse search, it is only necessary to be able to find, given any node n, the set of parent nodes of n such that there exists some valid operator from each of the parent nodes to n.
39.
For example, looking at a tree as a whole, one can talk about " the parent node " of a given node, but in general as a data structure a given node only contains the list of its children, but does not contain a reference to its parent ( if any ).
40.
Moreover, " C " must be nontrivial, because the trivial tree has " n " = 0 and " l " = 1 and is therefore not a counterexample . " C " therefore has at least one leaf whose parent node is an interior node.
