Refined Buneman Trees

Figure 6.7: An example a node which is a valid insertion node (green), and one
which is not a valid insertion node (red). A node is a valid insertion node if all
subtrees of that node contain only nodes from U (blue nodes) or from V (black
nodes), but not from both.
• w u = |U|∨w v < |V | — in this case, we know that the subtree of w given
by its parent edge only contains elements from V . If we assume that we
found w in a bottom–up traversal, and that w is the first node with the
property w u = |U|, w mustbethenodewearelookingfor.Itcannotbea
node below w, since that node would have a subtree with mixed u’s and
v’s, i.e. the subtree given by its parent edge.
• w u < |U|∨w v = |V | — similar argument as in the case w u = |U|∨w v < |V |.
• w u = |U|∨w v = |V | — in this case, the node we chose at randon to be
the intermediate root, must be the insertion node we are looking for.
Counting can be done in linear time since we only need to visit each node
once, and there are at most 2n − 2 nodes in an unrooted tree with n leaves.
Searching can afterwards be done in time O(n), visiting each node exactly once
and testing each node in constant time until we find the one we are looking for.
6.2.2 Delete
The delete operation is somewhat similar to the insert operation. Again we
start out by counting U’s and V ’s in a bottom-up fashion. The edge e we are
looking for has the property that its destination node is a node w, whereall
subtrees of w contain only elements from U or from V , but not both. In other
words, w isthenodewhereu w = |U| and v w = 0 or vice versa. And, once the
node w is found, the edge e has also been found, since we can keep track of both
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