Introduction
This document explains how to use the avl_tree template. Adelson-Velskii and Landis Balanced Binary Search Trees (or AVL Trees) are described in many good textbooks on fundamental data structures. The best web page I’ve been able to find on the topic is “A Visual Basic AVL Tree Container Class”.
This document, as well as the source code it describes, is in the public domain.
To avoid possible confusion about the terms that I use in this document (and in the source comments), here is a summary description of AVL Trees. An AVL Tree is a set of nodes (or elements). Each node is associated with a unique key value. The key values can be ordered from least to greatest. Each node in the tree may (or may not) have a less child node, and it may (or may not) have a greater child node. If node A is a child of node B, then B is the parent of A. If A is the less child of B, A’s key must be less than B’s key. Similarly, if A is the greater child of B, A’s key must be greater than B’s key. All nodes in a tree have exactly one parent, except for the root node, which has no parent. Node A is a descendant of node C if C is A’s parent, or if A’s parent is a descendant of C. If a node is not the root of the entire tree, it is the root of a subtree consisting of the node and all its descendants. The lesser subtree of a node is the subtree whose root is the less child of the node. The greater subtree of a node is the subtree whose root is the greater child of the node. The depth of a node is one more than the depth of its parent. The depth of the root node is 1. The depth of a tree is the maximum node depth. The balance factor of a node is the depth of its greater subtree minus the depth of its lesser subtree, with non-existent subtrees being considered to have a depth of 0. In an AVL tree, the balance factor of any node can only be -1, 0 or 1.
There are several open-source C and C++ implementations of AVL Trees available (see “Hot Links”, then “Data Structures” at C/C++ User’s Group Page ). But as far as I know, this is the only one that manipulates the nodes of the tree using abstract “handles” instead of concrete pointers. If all the nodes are in a single array, you can use node indexes as handles instead of node pointers. This approach makes it possible to compress the size of the nodes if memory is tight. Index handles can make tree persistence as simple as writing the node array out with a single disk write, and reading it back in with a single disk read. The template also allows for a tree to be in secondary storage, with nodes being “paged” in and out of memory.
To achieve the desired level of abstraction, the avl_tree template uses lots of short inline functions. Because of this, function inlining can significantly improve performance when using the template. If the test suite (test_avl.cpp) is compiled with GNU GCC using level 1 optimization (-O option), it executes twice as fast as when the test suite is compiled without optimization (the default).
The template code makes no use of recursion. The implementation is stack-friendly in general, except perhaps for the iter class. Instances of iter contain an array of handles whose dimension is the maximum tree depth minus one. Since key comparisons can potentially be complex, the code avoids repeated comparisons of the same pair of node key values. To avoid clutter, default destructor functions are not documented.
Source Files
- The source code for the template is in the header file avl_tree.h
- A test suite for the template is in the file test_avl.cpp
- avl_ex1.cpp shows an example instantiation of the template using pointers as handles
- avl_ex2.cpp shows an example instantiation of the template using array indexes as handles
All of this code compiles with a contemporary version of GNU GCC, and with Visual C++ .NET.
Reference Classes
To help describe the constraints on template class/typename parameters, or on member types of template class parameters, I like to use reference classes. This doesn’t necessary mean that the type being constrained has to use the reference class as its definition. It is only necessary that every possible usage of the reference class or one of its instances is also a possible usage of the constrained type or one of its instances. When an identifier with the prefix ANY_ is used, this means that all occurrences of that identifier should be substituted with the same type (or with types that implicitly convert to the substituted type). Take, for example, the function template:
template <class A>
void foo(A &a) { a.x(a.y()); }
The reference class for the parameter A would be:
class A
{
public:
void x(ANY_px p);
ANY_px y(void);
};
The following class could be passed as the class A parameter to the template:
struct someA
{
public:
static double x(int aintp);
signed char y(bool f = true) const;
};
Since the return type of x() is void in the reference class, it can return any type (or be void) in the actual parameter class. y() can return signed char because signed char implicitly converts to int . Member functions can be made static or const because these make the usage of a function more, not less, flexible.
Namespace
The avl_tree template is in the abstract_container namespace. The AVL Tree header file also defines this enumerated type:
enum search_type
{
EQUAL = 1,
LESS = 2,
GREATER = 4,
LESS_EQUAL = EQUAL | LESS,
GREATER_EQUAL = EQUAL | GREATER
};
in the abstract_container namespace.
Template Parameters
The avl_tree template begins with:
template <class abstractor, unsigned max_depth = 32>
class avl_tree . . .
Members of Reference Class for abstractor Template Parameter
All members of the reference class are public.
Type handle
Each node has to be associated with a node handle, which is a unique value of the handle type. Here is the reference class for handle:
class handle
{
public:
handle(void);
handle(handle &h);
void operator = (handle &h);
bool operator == (handle &h);
};
Type key
Each node has to be associated with a key, which is a unique value of the key type. The difference between a key and a handle is that a node can be conveniently “looked up” using its handle, but it can’t be conveniently looked up using its key. In fact, the whole point of this template is to make it convenient to look up a node given its key. Here is the reference class for key:
classkey
{
public:
key(key&k);
};
Type size
The type size is char, short, int or long, signed or unsigned. It must be large enough the hold the maximum possible number of nodes in the tree.
Functions get_less, get_greater
handle get_less(handle h, bool access = true);
handle get_greater(handle h, bool access = true);
Return the handle of the less/greater child of the node whose handle is h. If access is true, and the child node is in secondary storage, it has to be read into memory. If access is false, the child node does not have to be read into memory. Ignore the access parameter if your instantiation makes no use of secondary storage.
Functions set_less, set_greater
void set_less(handle h, handle lh);
void set_greater(handle h, handle gh);
Given the handle h of a node, set the handle of the less/greater child of the node.
Function get_balance_factor
int get_balance_factor(handle h);
Return the balance factor of the node whose handle is h.
Function set_balance_factor
void set_balance_factor(handle h, int bf);
Set the balance factor of the node whose handle is h. The only possible balance factor values are -1, 0 and 1.
Function compare_key_node
int compare_key_node(key k, handle h);
Compares a key with the key of a node. Returns a negative value if the key is less than the node’s key. Returns zero if the key is the same as the node’s key. Returns a positive value if the key is greater than the node’s key.
Function compare_node_node
int compare_node_node(handle h1, handle h2);
Compares the keys of two nodes. Returns a negative value if the first node’s key is less than the second node’s key. Returns zero if the first node’s key is the same as the second node’s key. Returns a positive value if the first node’s key is greater than the second node’s key.
Function null
handle null(void);
Always returns the same, invalid handle value, which is called the null value.
Function read_error
bool read_error(void);
Returns true if there was an error reading secondary storage. If your instantiation of the template makes no use of secondary storage, use this definition:
bool read_error(void) { return(false); }
Parameterless Constructor
abstractor(void);
max_depth
This is the maximum tree depth for an instance of the instantiated class. You almost certainly want to choose the maximum depth based on the maximum number of nodes that could possibly be in the tree instance at any given time. To do this, let the maximum depth be M such that:
MN(M) <= maximum number of nodes < MN(M + 1)
where MN(d) means the minimum number of nodes in an AVL Tree of depth d. Here is a table of MN(d) values for d from 2 to 45.
| D | MN(d) |
| 2 | 2 |
| 3 | 4 |
| 4 | 7 |
| 5 | 12 |
| 6 | 20 |
| 7 | 33 |
| 8 | 54 |
| 9 | 88 |
| 10 | 143 |
| 11 | 232 |
| 12 | 376 |
| 13 | 609 |
| 14 | 986 |
| 15 | 1,596 |
| 16 | 2,583 |
| 17 | 4,180 |
| 18 | 6,764 |
| 19 | 10,945 |
| 20 | 17,710 |
| 21 | 28,656 |
| 22 | 46,367 |
| 23 | 75,024 |
| 24 | 121,392 |
| 25 | 196,417 |
| 26 | 317,810 |
| 27 | 514,228 |
| 28 | 832,039 |
| 29 | 1,346,268 |
| 30 | 2,178,308 |
| 31 | 3,524,577 |
| 32 | 5,702,886 |
| 33 | 9,227,464 |
| 34 | 14,930,351 |
| 35 | 24,157,816 |
| 36 | 39,088,168 |
| 37 | 63,245,985 |
| 38 | 102,334,154 |
| 39 | 165,580,140 |
| 40 | 267,914,295 |
| 41 | 433,494,436 |
| 42 | 701,408,732 |
| 43 | 1,134,903,169 |
| 44 | 1,836,311,902 |
| 45 | 2,971,215,072 |
If, in a particular instantiation, the maximum number of nodes in a tree instance is 1,000,000, the maximum depth should be 28. You pick 28 because MN(28) is 832,039, which is less than or equal to 1,000,000, and MN(29) is 1,346,268, which is strictly greater than 1,000,000.
If you insert a node that would cause the tree to grow to a depth greater than the maximum you gave, the results are undefined.
Each increase of 1 in the value of max_depth increases the size of an instance of the iter class by sizeof(handle). The only other use of max_depth is as the size of bit arrays used at various places in the code. Generally, the number of bytes in a bit array is the size rounded up to a multiple of the number of bits in an int, and divided by the number of bits in a byte. All this is a roundabout way of saying that, if you don’t use iter instances, you can guiltlessly add a big safety margin to the value of max_depth.
Public Members
Type handle
Same as handle type member of the abstractor parameter class.
Type key
Same as key type member of the abstractor parameter class.
Type size
Same as size type member of the abstractor parameter class.
Function insert
handle insert(handle h);
Insert the node with the given handle into the tree. The node must be associated with a key value. The initial values of the node’s less/greater child handles and its balance factor are don’t-cares. If successful, this function returns the handle of the inserted node. If the node to insert has the same key value as a node that’s already in the tree, the insertion is not performed, and the handle of the node already in the tree is returned. Returns the null value if there is an error reading secondary storage. Calling this function invalidates all currently-existing instances of the iter class (that are iterating over this tree).
Function search
handle search(key k, search_type st = EQUAL);
Searches for a particular node in the tree, returning its handle if the node is found, and the null value if the node is not found. The node to search for depends on the value of the st parameter.
| Value of st | Node to search for |
EQUAL | Node whose key is equal to the key k. |
LESS | Node whose key is the maximum of the keys of all the nodes with keys less than the key k. |
GREATER | Node whose key is the minimum of the keys of all the nodes with keys greater than the key k. |
LESS_EQUAL | Node whose key is the maximum of the keys of all the nodes with keys less than or equal to the key k. |
GREATER_EQUAL | Node whose key is the minimum of the keys of all the nodes with keys greater than or equal to the key k. |
Function search_least
handle search_least(void);
Returns the handle of the node whose key is the minimum of the keys of all the nodes in the tree. Returns the null value if the tree is empty or an error occurs reading from secondary storage.
Function search_greatest
handle search_greatest(void);
Returns the handle of the node whose key is the maximum of the keys of all the nodes in the tree. Returns the null value if the tree is empty or an error occurs reading from secondary storage.
Function remove
handle remove(key k);
Removes the node with the given k from the tree. Returns the handle of the node removed. Returns the null value if there is no node in the tree with the given key, or an error occurs reading from secondary storage. Calling this function invalidates all currently-existing instances of the iter class (that are iterating over this tree).
Function purge
void purge(void);
Removes all nodes from the tree, making it empty.
Function is_empty
bool is_empty(void);
Returns true if the tree is empty.
Function read_error
void read_error(void);
Returns true if an error occurred while reading a node of the tree from secondary storage. When a read error has occurred, the tree is in an undefined state.
Parameterless Constructor
avl_tree(void);
Initializes the tree to the empty state.
Function Template build
template<typename fwd_iter>
bool build(fwd_iter p, size num_nodes);
Builds a tree from a sequence of nodes that are sorted in ascending order by their key values. The number of nodes in the sequence is given by num_nodes. p is a forward iterator that initially refers to the first node in the sequence. Here is the reference class for the fwd_iter:
class fwd_iter
{
public:
fwd_iter(fwd_iter &);
handle operator * (void);
void operator ++ (int);
};
Any nodes in the tree (prior to calling this function) are purged. The iterator will be incremented one last time when it refers to the last node in the sequence. build() returns false if a read error occurs while trying to build the tree. The time complexity of this function is O(n x log n), but it is more efficient than inserting the nodes in the sequence one at a time, and the resulting tree will generally have better balance.
Copy Constructor and Assignment Operator?
If the abstractor class has a copy constructor and assignment operator, the avl_tree instantiation will have a (default) copy constructor and assignment operator.
Class iter
Instances of this member class are bi-directional iterators over the ascendingly sorted (by key) sequence of nodes in a tree. The subsections of this section describe the public members of iter.
-
Parameterless Constructor
iter(void);
Initializes the iterator to the null state.
-
Function start_iter
void start_iter(avl_tree &tree, key k, search_type st = EQUAL);
Causes the iterator to refer to a particular node in the tree that is specified as the first parameter. If the particular node cannot be found in the tree, or if a read error occurs, the iterator is put into the null state. The particular node to refer to is determined by the st parameter.
| Value of st | Node to search for |
EQUAL | Node whose key is equal to the key k. |
LESS | Node whose key is the maximum of the keys of all the nodes with keys less than the key k. |
GREATER | Node whose key is the minimum of the keys of all the nodes with keys greater than the key k. |
LESS_EQUAL | Node whose key is the maximum of the keys of all the nodes with keys less than or equal to the key k. |
GREATER_EQUAL | Node whose key is the minimum of the keys of all the nodes with keys greater than or equal to the key k. |
-
Function start_iter_least
void start_iter_least(avl_tree &tree);
Cause the iterator to refer to the node with the minimum key in the given tree. Puts the iterator into the null state if the tree is empty or a read error occurs.
-
Function start_iter_greatest
void start_iter_greatest(avl_tree &tree);
Cause the iterator to refer to the node with the maximum key in the given tree. Puts the iterator into the null state if the tree is empty or a read error occurs.
-
Operator *
handle operator * (void);
Returns the handle of the node that the iterator refers to. Returns the null value if the iterator is in the null state.
-
Prefix and Postfix Operator ++
void operator ++ (void);
void operator ++ (int);
Causes the iterator to refer to the node whose key is the next highest after the key of the node the iterator currently refers to. Puts the iterator into the null state if the key of the node currently referred to is the maximum of the keys of all the nodes in the tree, or if a read error occurs. Has no effect if the iterator is already in the null state.
-
Prefix and Postfix Operator --
void operator -- (void);
void operator -- (int);
Causes the iterator to refer to the node whose key is the next lowest after the key of the node the iterator currently refers to. Puts the iterator into the null state if the key of the node currently referred to is the minimum of the keys of all the nodes in the tree, or if a read error occurs. Has no effect if the iterator is already in the null state.
-
Function read_error
Returns true if a read error occurred.
-
Default Copy Constructor and Assignment Operator
These member functions exist and can be safely used.
Protected Members
Variable abs
abstractor abs;
Variable root
handle root;
Contains the handle of the root node of the AVL Tree. Contains null value if the tree is empty.
Other Protected Members
The other protected members are most easily understood by reading the source code.
History
- 12th September, 2002 - Fixed some problems in the code for handling errors if tree nodes are in secondary strorage
