Handling underflow using transfer

Deleting the last entry from a node

Deleting an entry e from a node p with exactly 1 entry:

Comment:

The exact meaning of underflow is:

You will see that delete will also propagate up into the tree by a merge operation

The merge operation will remove:
The resulting node in a 2,4-tree will have:

So technically, the node is not empty, but underflow because:

There are 2 methods used to handle underflow (= too few subtrees):

Transfer one subtree from an immediate sibling node into the underflow node
Merge the underflow node (with has < 2 subtrees) with one of the immediate sibling nodes
(After the merge, the resulting node will have enough subtrees)

Overview: handling underflow

Brief overview for handling an underflow situation in the node p (i.e., node p has fewer than 2 subtrees):

If one (left or right) immediate sibling node of node has ≥ 3 subtrees:

Transfer a subtree from the sibling node into the underflow node

Because a subtree requires a search key to decide the branch direction, to keep the correct key storage structure, the tree transfer operation requires:

Transfer the entry from node p's parent to replace the deleted entry e in node p, and
Transfer an entry from the immediately sibling to node p's parent

If both immediate sibling nodes of node p have 1 entry:

Merge:
into one single node

Notice that:

The merge operation will remove:

Consequently:

Node p's parent node can itself become underflow (i.e., has 1 subtree) !!!

Solution:

Repeat the same procedure at p's parent node
In other words:
In yet other words:

This brief overview is probably incomprehensive right now...
That's OK. You can skip it for now.
Come back and read it again after we discuss the techniques in details.
Because the above description contains certain insights that help you understand the details.

Clarification: the "immediate" sibling node

Terminology:

Sibling nodes are nodes that share the same parent node
Example:
What I (not in text books) call immediate sibling nodes, are sibling nodes that are next to each other
Example:

How to find the immediate sibling nodes:

The immediate right sibling node of a node p is found at the root of the "next right" subtree of p's parent node
Example:
The immediate left sibling node of a node p is found at the root of the "last left" subtree of p's parent node
Example:

Solving underflow by transfering with a sibling that has ≥ 3 subtrees

Technique:

If an underflow condition occurs in a delete operation, and one of the immediate sibling nodes has ≥ 3 subtrees, we can solve the underflow condition by:

Transfer with its right sibling:

Let's start with a delete operation that caused an underflow: remove(6)
Transfer a subtree from the right sibling:
In order to provide the correct search direction for the transfered subtree, we must also transfer so entries (with keys)
Spefically:
Graphically:
More complete picture:

Transfer with its left sibling:

Delete the entry:

Transfer a subtree from your left sibling:

To support the search structure, we must also transfer some entries as follows:

Note:

Again, we must transfer the entry from the parent node into the underflow flow (along with the transfered subtree) to keep the search structure consistent.
We must also transfer an entry from the left child into the parent node

Very important note:

When we transfer a subtree reference into another node, we must:

Example:

We will see later that the delete operation can cause cascaded transfers up the tree.
We may have to perform a transfer where the red reference is a reference to a subtree:
After the transfer operation, the parent node of the subtree has changed:
You must update the parent reference of the root node in the subtree !!!