A *binary tree* is often defined recursively as either
(a) being empty, or (b) consisting of a root node together
with left and right subtrees, both of which are binary
trees.
It is this definition that is most useful from the point of
view of data structures in computer science.
From the mathematical point of view these objects are not
trees.
Changing (a) from "empty" to "a single vertex" gives a
definition equivalent to that of ordered trees in which
every node is either a leaf (no children) or has two
children.
These are sometimes called *extended binary trees*.
In either case, they are usually parameterized by *n*,
the number of applications of rule (b) that are used.

In the extended binary tree illustrated above there are
5 *internal nodes* (the brown
circles, which correspond to applications of rule (b))
and 6 *leaves* (the blue squares, which correspond to
applications of rule (a)).
By removing the blue edges and the leaves you obtain the
corresponding binary tree.

The number of extended binary trees with *n* internal nodes
is given by the *n*th Catalan number
(2*n*)!/[(*n*+1)!*n*!]
which, for *n* = 1,2,...,15, has the values
1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,
2674440, 9694854.
This is sequence
**A000100**(M1459) in
Neil J. Sloane's
database
of integer sequences.

The number of extended binary trees with *n* internal nodes
and leftmost leaf at level *m* is
*m*(2*n*-*m*-1)!/[(*n*-*m*)!*n*!].
The table of these numbers, shown below, is known as
"Catalan's triangle" if the rows are read backwards.
This is sequence
**A033184** in
Neil J. Sloane's
database
of integer sequences.

m = | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

n=1
| 1 | |||||||

2 | 1 | 1 | ||||||

3 | 2 | 2 | 1 | |||||

4 | 5 | 5 | 3 | 1 | ||||

5 | 14 | 14 | 9 | 4 | 1 | |||

6 | 42 | 42 | 28 | 14 | 5 | 1 | ||

7 | 132 | 132 | 90 | 48 | 20 | 6 | 1 | |

8 | 429 | 429 | 297 | 165 | 75 | 27 | 7 | 1 |

Binary trees with *n* nodes are often represented as bitstrings
b_{1} b_{2} ... b_{2n} obtained by traversing
the equivalent extended binary tree in preorder,
recording a 1 as each internal node is encountered, and a 0
each time a leaf is encountered, except for the final leaf,
which is ignored.
The position of the leftmost 0 is one greater than the level of the
leftmost leaf (with the root at level 0).
Thus for the tree illustrated above the bitstring is **1101001010**.
Every such bitstring corresponds to a well-formed parentheses string,
obtained by regarding 1 a left paren and 0 as a right paren.
Our bitstring is transformed into ** (()())()()**.
If "alternate output" is selected, then the bitstring is transformed into
balls, black for an 1 and white for a 0.
For our example, the output is
.

If Gray code order is selected then each tree sequence differs from its predecessor by the transposition of two bits. It is the position of these two bits that are output if Gray code output is selected.

If generation by rotations is selected, then each successive tree differs by a rotation from its predecessor. A rotation is illustrated in the animation on the left, which alternately rotates the tree about the blue edge; left, right, left, right, etc. The relative order of the three subtrees is maintained but the root of the tree changes as illustrated. Rotations are used to maintain balance in binary search trees. There is an interesting correspondence between rotations and diagonal flips in triangulations of convex polygons.

An *ordered tree* is a rooted tree in which the relative order of
subtrees matters.
An *ordered forest* is an ordered collection of ordered trees.
There is a one-to-one correspondence between ordered forests with
*n* nodes and binary trees with *n* nodes.
This correspondence is best explained using the well-formed parentheses strings
with *n* left and *n* right parentheses. If the *n* left parentheses and
*n* right parentheses are labelled *1* through *n* from left to right in the string,
then there are *n* pairs of matching left/right pairs of parentheses. These pairs
correspond to the nodes of the corresponding ordered forest. Listing the pairs
in increasing order of the left parentheses corresponds to a preorder listing of
the nodes of the forest and listing the pairs in increasing order of the right
parentheses corresponds to a postorder listing of the nodes of the forest.

For the parentheses string mentioned above, ** (()())()()**, the pairs of
matching left/right pairs are

Thus the algorithms for generating binary trees may also be thought of as algorithms for generating ordered forests.

Related Links:

- Red-Black Trees, a type of binary tree.

The lex order algorithm was developed independently by several authors and is periodically re-discovered! The Gray code algorithm used is from the paper: F.Ruskey and A.Proskurowski, Generating Binary Trees by Transpositions, J.Algorithms, 11 (1990) 68-84. The algorithm for generating binary trees by rotations is from the paper: J. Lucas, D. Roelants van Baronaigien, and F. Ruskey, On Rotations and the Generation of Binary Trees, J. Algorithms, 15 (1993) 343-366.

Programs available:

- Lexicographic order: (C program) (Pascal program)
- Gray code order: (C program) (Pascal program)

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It was last modified on Tuesday, 24-May-2011 10:50:23 PDT.

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©Frank Ruskey, 1995-2003.

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