Leaf-to-Leaf Distances in Catalan Tree Graphs
Andrew M. Goldsborough, Jonathan M. Fellows, S. Alex Rautu, Matthew Bates,
George Rowlands, and Rudolf A. Römer
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, UK, CV4 7AL
Leg Depths
Motivation
Path Lengths
• Two-point correlation functions in tensor networks are related to the path length
through the network that connects the two sites. [1,2]
• Tensor network strong-disorder renormalization group (tSDRG) [3] constructs a
binary tree tensor network (TTN).
• Correlation functions for tSDRG need to be averaged over separation (r) and over
many disorder realizations.
• Can we find an analytic expression for the average distance for a given separation
• In order to find the distances it is necessary to find the average depth of a leg with xcoordinate, or penetration, m.
• Let Pn,r be the total leaf-to-leaf distance when summed over all pairs of leaves
separated by r in all trees with n vertices.
• Let the sum of the depths of leg m in the set of trees with n vertices be Dm,n.
• Let the average distance be:
• The average depth of leg m is therefore:
for an idealized set of trees?
• It can be shown that the the sum of the path lengths is:
Definitions
Vertex
• An equation for Dm,n can be found by looking at a box decomposition again:
Root node
=
, ... ,
...
...
, ... ,
, ... ... ,
...
...
...
• The generating function is the same as for the sum of the depths, hence:
• The contribution to the depths are one for the root node of each graph and the
depth in each sub-tree multiplied by the degeneracy of the opposing sub-tree:
Leaf
x
• Thus the average path length for separation r is equal to the depth of leg r:
• Asymptotic (large n) properties can be found using Stirling's approximation:
Separation r
• The generating function for Dm,n will take the form:
• A binary tree is a connected, 3-valent graph with no loops.
• The Catalan numbers become:
• The root node is the vertex with two degrees that defines the top of the graph. Then
the rest of the vertices have two daughter nodes and one parent.
• A leaf node has no daughters, also referred to as a leg.
• The depth of a leaf is the number of vertices from the root node to the leaf.
• Manipulation of the generating function obtains a solution for Dm,n :
• Average path length when n → ∞ is:
• We enumerate the leaves from left to right to imitate a lattice. We can then define
a separation as r = |x1 – x2|.
• Leaf-to-leaf distance or leaf-to-leaf path length is the number of vertices connecting
two leaves.
• Using various identities, this can be simplified to a closed form:
An Exercise for the Reader
• Prove the identity:
Binary Catalan Trees
• An alternative form of the generating function is:
• Binary Catalan trees are the set of unique binary tree graphs with n vertices. [4]
• The number of unique trees with n vertices is given by Catalan number Cn. [5] This
can be seen by looking at how the trees decompose:
Generating Functions
• Generating functions are a versatile method for finding properties of sequences and
• Generating functions have the following properties:
series. [6]
• Given a general infinite series (a0, a1, a2...) it is always possible to define a function:
• Write this as a box-decomposition:
• The n-th term of the series is then the coefficient of x n in the expansion of the
=
, ... ,
,
...
...
...
,
...
...
generating function:
• The generating function for the Catalan numbers is:
=
...
...
• Summing the set of trees gives Segner's recurrence formula [4]:
Conclusions
• The Catalan numbers can also be expressed in terms of binomials: and factorials:
• We have analysed average leaf depths and leaf-to-leaf distances in Catalan trees, extending
the work in [2].
Contact Details
• Email: a.goldsborough@warwick.ac.uk, r.roemer@warwick.ac.uk
• Website: http://www.warwick.ac.uk/andrewgoldsborough/
• We derived an analytic form for the average depth of a leaf penetration m by using
• The first few Catalan numbers are:
generating functions.
• We showed that the average leaf-to-leaf path distance with separation r is equal to the
average depth of leaf with penetration r.
References
[1] G. Evenbly and G. Vidal, J. Stat. Phys. 145, 891 (2011).
[2] A. M. Goldsborough, S. A. Rautu, and R. A. Römer, arXiv:1406.4079 (2014).
[3] A. M. Goldsborough, and R. A. Römer, Phys. Rev. B, 89, 214203 (2014).
[4] R. Sedgewick, and P. Flajolet, An Introduction to the Analysis of Algorithms, 2nd Edition. Addison-Wesley (2013).
[5] T. Koshy, Catalan Numbers with Applications. Oxford University Press (2009).
[6] H. S. Wilf, Generatingfunctionology, 2nd Edition. Academic Press (1994).