Chapter 5 |

**1.**
In a BST, the time for an insertion or deletion is the time required
to visit each node from the root down to the node of interest, plus
some time to perform the operation itself. Functions *bst_probe*()
and *bst_delete*() contain only a single loop each, which iterates
once for each node examined. As the tree grows, the time for the
actual operation loses significance and the total time for the
operation becomes essentially proportional to the height of the tree,
which is approximately *log2* (*n*) in the best case
(see Analysis of AVL Balancing Rule).

We were given that the additional work for rebalancing an AVL or
red-black tree is at most a constant amount multiplied by the height
of the tree. Furthermore, the maximum height of an AVL tree is 1.44
times the maximum height for the corresponding perfectly balanced
binary tree, and a red-black tree has a similar bound on its height.
Therefore, for trees with many nodes, the worst-case time required to
insert or delete an item in a balanced tree is a constant multiple of
the time required for the same operation on an unbalanced BST in the
best case. In the formal terms of computer science, insertion and
deletion in a balanced tree are O(*log* *n*) operations, where *n* is
the number of nodes in the tree.

In practice, operations on balanced trees of reasonable size are, at worst, not much slower than operations on unbalanced binary trees and, at best, much faster.

**1.**
Variable *y* is only modified within <Step 1: Search AVL tree for insertion point 150>. If *y* is set during the loop, it is set to *p*,
which is always a non-null pointer within the loop. So *y* can only be
`NULL` if it is last set before the loop begins. If that is true, it
will be `NULL` only if *tree*->*avl_root* == `NULL`. So, variable *y* can
only be `NULL` if the AVL tree was empty before the insertion.

A `NULL` value for *y* is a special case because later code assumes that
*y* points to a node.

**1.**
No. Suppose that *n* is the new node, that *p* is its parent, and that
*p* has a - balance factor before *n*'s insertion (a similar argument
applies if *p*'s balance factor is +). Then, for *n*'s insertion to
decrease *p*'s balance factor to -2, *n* would have to be the left
child of *p*. But if *p* had a - balance factor before the insertion,
it already had a left child, so *n* cannot be the new left of *p*. This
is a contradiction, so case 3 will never be applied to the parent of a
newly inserted node.

In the leftmost tree, case 2 applies to the root's left child and the root's balance factor does not change. In the middle tree, case 1 applies to the root's left child and case 2 applies to the root. In the rightmost tree, case 1 applies to the root's left child and case 3 applies to the root. The tree on the right requires rebalancing, and the others do not.

**3.**
Type **char** may be signed or unsigned, depending on the C compiler
and/or how the C compiler is run. Also, a common use for subscripting
an array with a character type is to translate an arbitrary character
to another character or a set of properties. For example, this is a
common way to implement the standard C functions from *ctype*.*h*. This
means that subscripting such an array with a **char** value can have
different behavior when **char** changes between signed and unsigned
with different compilers (or with the same compiler invoked with
different options).

**See also:**
[ISO 1990], section 6.1.2.5;
[Kernighan 1988], section A4.2.

652. <Step 3: Update balance factors after AVL insertion, with bitmasks 652> =for(p=y;p!=n;p=p->avl_link[cache& 1],cache>>= 1)if((cache& 1) == 0)p->avl_balance--;else

p->avl_balance++;

Also, replace the declarations of *da*[] and *k* by these:

unsignedlongcache= 0; /* Cached comparison results. */intk= 0; /* Number of cached comparison results. */

and replace the second paragraph of code within the loop in step 1 by this:

if(p->avl_balance!= 0)z=q,y=p,cache= 0,k= 0;dir=cmp> 0;if(dir)cache|= 1ul<<k;k++;

It is interesting to note that the speed difference between this version and the standard version was found to be negligible, when compiled with full optimization under GCC (both 2.95.4 and 3.0.3) on x86.

**1.**
Because then *y*'s right subtree would have height 1, so there's no
way that *y* could have a +2 balance factor.

**2.**
The value of *y* is set during the search for *item* to point to the
closest node above the insertion point that has a nonzero balance
factor, so any node below *y* along this search path, including *x*,
must have had a 0 balance factor originally. All such nodes are updated
to have a nonzero balance factor later, during step 3. So *x* must have
either a - or + balance factor at the time of rebalancing.

**3.2.**

**3.3.**

**4.**
*w* should replace *y* as the left or right child of *z*. *y* != *z*->*avl_link*[0] has the value 1 if *y* is the right child of *z*, or
0 if *y* is the left child. So the overall expression replaces *y*
with *w* as a child of *z*.

The suggested substitution is a poor choice because if *z* == (**struct** **avl_node** *) &*tree*->*root*, *z*->*avl_link*[1] is undefined.

**1.**
This approach cannot be used in libavl (see Exercise 4.8-3).

653. <Case 3 in AVL deletion, alternate version 653> =structavl_node*s;da[k] = 1;pa[k++] =p;for(;;)

{da[k] = 0;pa[k++] =r;s=r->avl_link[0];if(s->avl_link[0] ==NULL)break;r=s; }p->avl_data=s->avl_data;r->avl_link[0] =s->avl_link[1];p=s;

**2.**
We could, if we use the standard libavl code for deletion case 3.
The alternate version in Exercise 1 modifies item
data, which would cause the wrong value to be returned later.

**1.**
Tree *y* started out with a + balance factor, meaning that its right
subtree is taller than its left. So, even if *y*'s left subtree had
height 0, its right subtree has at least height 1, meaning that *y*
must have at least one right child.

**2.**
Rebalancing is required at each level if, at every level of the tree,
the deletion causes a +2 or -2 balance factor at a node *p* while
there is a +1 or -1 balance factor at *p*'s child opposite the
deletion.

For example, consider the AVL tree below:

Deletion of node 32 in this tree leads to a -2 balance factor on the left side of node 31, causing a right rotation at node 31. This shortens the right subtree of node 28, causing it to have a -2 balance factor, leading to a right rotation there. This shortens the right subtree of node 20, causing it to have a -2 balance factor, forcing a right rotation there, too. Here is the final tree:

Incidentally, our original tree was an example of a “Fibonacci
tree”, a kind of binary tree whose form is defined recursively, as
follows. A Fibonacci tree of order 0 is an empty tree and a Fibonacci
tree of order 1 is a single node. A Fibonacci tree of order *n* >= 2
is a node whose left subtree is a Fibonacci tree of order *n* - 1 and
whose right subtree is a Fibonacci tree of order *n* - 2. Our example
is a Fibonacci tree of order 7. Any big-enough Fibonacci tree will
exhibit this pathological behavior upon AVL deletion of its maximum
node.

**1.**
At this point in the code, *p* points to the *avl_data* member of an
**struct** **avl_node**. We want a pointer to the **struct** **avl_node** itself.
To do this, we just subtract the offset of the *avl_data* member within
the structure. A cast to **char** * is necessary before the subtraction,
because *offsetof* returns a count of bytes, and a cast to **struct** **avl_node** * afterward, to make the result the right type.