Namespaces
Variants
Actions

Difference between revisions of "Binary tree"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
A (planar) rooted tree for which every node has a left child, a right child, neither, or both. Three examples are:
 
A (planar) rooted tree for which every node has a left child, a right child, neither, or both. Three examples are:
  
Line 7: Line 8:
 
These three are all different.
 
These three are all different.
  
The number of binary trees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105301.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105302.png" /> left children, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105303.png" /> right children (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105304.png" />) is
+
The number of binary trees with $n$ nodes, $p$ left children, $q$ right children ($p+q=n-1$) is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105305.png" /></td> </tr></table>
+
$$\frac1n\binom np\binom{n}{p+1}=\frac1n\binom np\binom nq.$$
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105306.png" /> are called Runyon numbers or Narayama numbers.
+
The numbers $n^{-1}\binom np\binom{n}{p+1}$ are called Runyon numbers or Narayama numbers.
  
 
A complete binary tree is one in which every node has both left and right children or is a leaf (i.e., has no children). E.g., there are two complete binary trees with five nodes:
 
A complete binary tree is one in which every node has both left and right children or is a leaf (i.e., has no children). E.g., there are two complete binary trees with five nodes:
Line 19: Line 20:
 
Figure: b110530b
 
Figure: b110530b
  
A complete binary tree has an odd number of nodes, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105307.png" />, and then the number of leaves is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105308.png" />. Label the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b1105309.png" /> leaves from left to right with symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053010.png" />. Then the various complete binary trees with their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053011.png" /> leaves labelled in this order precisely correspond to all the different ways of putting brackets in the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053012.png" />, each way corresponding to a computation of the product by successive multiplications of precisely two factors each time. The number of ways of doing this, and hence the number of binary trees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053013.png" /> nodes, is the Catalan number
+
A complete binary tree has an odd number of nodes, say $2k+1$, and then the number of leaves is $k+1$. Label the $k+1$ leaves from left to right with symbols $x_1,\ldots,x_{k+1}$. Then the various complete binary trees with their $k+1$ leaves labelled in this order precisely correspond to all the different ways of putting brackets in the word $x_1\ldots x_{k+1}$, each way corresponding to a computation of the product by successive multiplications of precisely two factors each time. The number of ways of doing this, and hence the number of binary trees with $k+1$ nodes, is the Catalan number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053014.png" /></td> </tr></table>
+
$$\frac{1}{k+1}\binom{2k}{k},k=1,2,\ldots.$$
  
 
The problem of all such bracketings of a product (of numbers) was considered by E. Catalan in 1838 [[#References|[a1]]].
 
The problem of all such bracketings of a product (of numbers) was considered by E. Catalan in 1838 [[#References|[a1]]].
  
The correspondence between complete binary trees and (complete) bracketings gives a bijection between complete binary trees with leaves labelled with elements from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053015.png" /> and the [[Free magma|free magma]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110530/b11053016.png" />.
+
The correspondence between complete binary trees and (complete) bracketings gives a bijection between complete binary trees with leaves labelled with elements from a set $X$ and the [[Free magma|free magma]] on $X$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Catalan,  "Note sur une équation aux différences finies"  ''J. Math. Pures Appl.'' , '''3'''  (1838)  pp. 508–516</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Comtet,  "Advanced combinatorics" , Reidel  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Gessel,  R.P. Stanley,  "Algebraic enumeration"  R.L. Graham (ed.)  M. Grötschel (ed.)  L. Lovacz (ed.) , ''Handbook of Combinatorics'' , '''II''' , Elsevier  (1995)  pp. 1021–1062</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.P. Stanley,  "Enumerative combinatorics" , Wadsworth and Brooks/Cole  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Catalan,  "Note sur une équation aux différences finies"  ''J. Math. Pures Appl.'' , '''3'''  (1838)  pp. 508–516</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Comtet,  "Advanced combinatorics" , Reidel  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Gessel,  R.P. Stanley,  "Algebraic enumeration"  R.L. Graham (ed.)  M. Grötschel (ed.)  L. Lovacz (ed.) , ''Handbook of Combinatorics'' , '''II''' , Elsevier  (1995)  pp. 1021–1062</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.P. Stanley,  "Enumerative combinatorics" , Wadsworth and Brooks/Cole  (1986)</TD></TR></table>

Revision as of 10:40, 12 April 2014

A (planar) rooted tree for which every node has a left child, a right child, neither, or both. Three examples are:

Figure: b110530a

These three are all different.

The number of binary trees with $n$ nodes, $p$ left children, $q$ right children ($p+q=n-1$) is

$$\frac1n\binom np\binom{n}{p+1}=\frac1n\binom np\binom nq.$$

The numbers $n^{-1}\binom np\binom{n}{p+1}$ are called Runyon numbers or Narayama numbers.

A complete binary tree is one in which every node has both left and right children or is a leaf (i.e., has no children). E.g., there are two complete binary trees with five nodes:

Figure: b110530b

A complete binary tree has an odd number of nodes, say $2k+1$, and then the number of leaves is $k+1$. Label the $k+1$ leaves from left to right with symbols $x_1,\ldots,x_{k+1}$. Then the various complete binary trees with their $k+1$ leaves labelled in this order precisely correspond to all the different ways of putting brackets in the word $x_1\ldots x_{k+1}$, each way corresponding to a computation of the product by successive multiplications of precisely two factors each time. The number of ways of doing this, and hence the number of binary trees with $k+1$ nodes, is the Catalan number

$$\frac{1}{k+1}\binom{2k}{k},k=1,2,\ldots.$$

The problem of all such bracketings of a product (of numbers) was considered by E. Catalan in 1838 [a1].

The correspondence between complete binary trees and (complete) bracketings gives a bijection between complete binary trees with leaves labelled with elements from a set $X$ and the free magma on $X$.

References

[a1] E. Catalan, "Note sur une équation aux différences finies" J. Math. Pures Appl. , 3 (1838) pp. 508–516
[a2] L. Comtet, "Advanced combinatorics" , Reidel (1974)
[a3] I.M. Gessel, R.P. Stanley, "Algebraic enumeration" R.L. Graham (ed.) M. Grötschel (ed.) L. Lovacz (ed.) , Handbook of Combinatorics , II , Elsevier (1995) pp. 1021–1062
[a4] R.P. Stanley, "Enumerative combinatorics" , Wadsworth and Brooks/Cole (1986)
How to Cite This Entry:
Binary tree. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Binary_tree&oldid=31607
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article