The Kitchen Sink and Other Oddities

Atabey Kaygun

Number of isomorphism classes of binary trees

Description of the problem

Today, I would like to write a recursive integer function which returns the number of isomorphism classes of binary trees on \(N\) vertices.

Usually, binary trees are ordered in the sense that left and the right legs and the parent node of a node are distinguished. However, from a graph theoretical point of view one cannot make such distinctions. For example, the following trees are isomorphic and should not be treated different:

A solution

One solution is to order the legs of a binary tree at a leaf using the number of nodes. Then the number of isomorphism classes of binary trees with \(n\) nodes is \[ \Upsilon(n) = \sum_{i=1}^{\lfloor n/2\rfloor} \Upsilon(i)\Upsilon(n-i) \] This sequence is known as the half Catalan number sequence.

The code

I will implement the function using memoization. One easy simplification: there are no binary trees on an odd number of vertices. So, the function should return 0 in those cases.

(let ((trees-table (make-hash-table)))
  (defun count-trees (n)
    (cond ((< n 2) 0)
          ((= n 2) 1)
          ((oddp n) 0)
          (t (or (gethash n trees-table)
                 (setf (gethash n trees-table)
                       (loop for i from 2 to (truncate (/ n 2)) by 2 sum 
                                (* (count-trees i)
                                   (count-trees (- n i))))))))))


COUNT-TREES

Let us test some small cases:

(count-trees 4)
1

(count-trees 10)
3

and something large

(time (count-trees 8000))

  Evaluation took:
  36.029 seconds of real time
  35.990000 seconds of total run time (35.833000 user, 0.157000 system)
  [ Run times consist of 0.096 seconds GC time, and 35.894 seconds non-GC time. ]
  99.89% CPU
  100,629,095,448 processor cycles
  4,087,578,560 bytes consed

  3232141093235745665344768253734720516
  7342297504444684529062661316231995608
  4038515824845546492953132261903137938
  7966010529079271003447972790652241578
  8206406816856071412065687354798901876
  0649219370141965820051113589805464975
  2397963481179106703504826446504878676
  0392385535238326313673032176829783556
  3602543190380074041880499185199619335
  7945207827045652603373499406175098715
  8184355208213350952487290628925607546
  3707906599107858716044228169511202997
  6858926637293235940612562487817298830
  3079081302321473052588875555573601569
  3529216065337436102878546847455040515
  0610566567346071430324157979443528748
  1776874003522289975254482716357855448
  2749720193426413795851312059670041280
  1210558028720726955444344153850844642
  6840069864315651973661978689685755985
  4086589472218362086348088066853264040
  6090332927129152972240579076174395947
  1590857474894224251682285788864313728
  5252056876478568114443509291891942217
  5435984810655453096826038929095339647
  0148259035101364662214637068682039906
  3250011473176322945990682797100164933
  5757962253408591333972774878987447961
  8953645896600813795240150881524942335
  3435984852982717944789730584443462479
  7778544697328463515558487074753445601
  9380057954555865565491241402630290339
  1902640767792958026936655677896050611
  9605225459764871667242462221315141473
  0903283661137011449891228190093894579
  0178313713048384412092261220103786640
  0090906811373279976542110078795323972
  9803274495187972041661987750039103684
  2528641806248555800573142650408382033
  2767683734303099322684114018750141629
  7072684919777039651181132245538061092
  7208010182054524903979118242578421430
  77064916800810685342775787

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