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### 9.5.4 Step 4: Rebalance

Rebalancing after deletion in a TAVL tree divides into three cases. The first of these is analogous to case 1 in unthreaded AVL deletion, the other two to case 2 (see section 8.6 Insertion). The cases are distinguished, as usual, based on the balance factor of right child x of the node y at which rebalancing occurs:

 ```struct tavl_node *x = y->tavl_link[1]; assert (x != NULL); if (x->tavl_balance == -1) { <@xref{\NODE\, , Rebalance for -.> balance factor after TAVL deletion in left subtree,320} } else { q->tavl_link[dir] = x; if (x->tavl_balance == 0) { <@xref{\NODE\, , Rebalance for 0 balance factor after TAVL deletion in left subtree.>,321} break; } else /* x->tavl_balance == +1 */ { <@xref{\NODE\, , Rebalance for +.> balance factor after TAVL deletion in left subtree,322} } } ```
This code is included in @refalso{318

#### Case 1: x has - balance factor

This case is just like case 2 in TAVL insertion. In fact, we can even reuse the code:

 ```struct tavl_node *w; <@xref{\NODE\, , Rebalance for -.> balance factor in TAVL insertion in right subtree,310} q->tavl_link[dir] = w; ```
This code is included in @refalso{319

#### Case 2: x has 0 balance factor

If x has a 0 balance factor, then we perform a left rotation at y. The transformation looks like this, with subtree heights listed under their labels:

Subtree b is taller than subtree a, so even if h takes its minimum value of 1, then subtree b has height h == 1 and, therefore, it must contain at least one node and there is no need to do any checking for threads. The code is simple:

 ```y->tavl_link[1] = x->tavl_link[0]; x->tavl_link[0] = y; x->tavl_balance = -1; y->tavl_balance = +1; ```
This code is included in @refalso{319

#### Case 3: x has + balance factor

If x has a + balance factor, we perform a left rotation at y, same as for case 2, and the transformation looks like this:

One difference from case 2 is in the resulting balance factors. The other is that if h == 1, then subtrees a and b have height h - 1 == 0, so a and b may actually be threads. In that case, the transformation must be done this way:

This code handles both possibilities:

 ```if (x->tavl_tag[0] == TAVL_CHILD) y->tavl_link[1] = x->tavl_link[0]; else { y->tavl_tag[1] = TAVL_THREAD; x->tavl_tag[0] = TAVL_CHILD; } x->tavl_link[0] = y; y->tavl_balance = x->tavl_balance = 0; ```
This code is included in @refalso{319

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