Functional trees

Recently, a group of coworkers and me have published a book on various themes around development very much in the style of the famous 97 Things Every Programmer Should Know (available as a wiki) but in Spanish and with longer chapters.

One of the chapters accounts for a programming exercise that was proposed just before last year Xmas. We were asked to create a tree with zeroes and a star on the top as follows:


Dozens of solutions were proposed in a very short time and employing almost a programming language per submission. Most of the programs were written by my colleagues with a code-golf mentality, trying to trade readability for size and brag about their languages of choice. Of course, some of the programs had more exotic purposes like laughing about PowerPoint programming, the verbosity of enterprise software or demonstrating MSX assembler among other nerdy stuff.

We can use this small and simple problem to compare the imperative and functional solutions for the case in which the tree is parameterized with n, the number of stores or layers or zeroes.

As an example of the imperative approach I reproduce here the submission in C++ of Luis Osa, a close coworker, edited to accept the number of tree stores as argument (see the original on the book):

#include <iostream>
#include <string>
#include <stdlib.h>

int main(int argc, char **argv)
    int n = atoi(argv[1]);
    std::cout << std::string(n, ' ') << "*" << std::endl;
    for (int i = 0; i < n; i++) {
        std::cout << std::string(n - i, ' ')
                  << std::string(2 * i + 1, '0')
                  << std::endl;
    return 0;

Three main characteristics can be highlighted:

  • Effectful computation. The tree is directly written to the standard output line by line. That is a side effect, as the returned value is 0.
  • Looping. Every tree line but the first is printed by a loop that structures the whole computation.
  • Monolithic logic. This simple program is structured as single monolithic function as it is difficult to extract worthy auxiliary functions.

So far, so good and more readable than the shortest submission in GolfScript (" "9*"*"9,{n\.4$>\.+)"0"*}%) but if this wasn’t fire-and-forget code we would be worried about testability and modularity for the long-term readability. The functional style can help here by the following means:

  • Referential transparency. In FP style, the tree function would return a sequence of tree lines instead of printing it directly as transparent functions can only compute results from their inputs and no other thing. This will save the day for the testing at the price of constructing the necessary datastructures.
  • Higher order functions. Low level looping is not necessary as the standard library of functions (map, reduce et al.) provide all the “looping” patterns to be parameterized with expressions.
  • Modular design. Both referential transparency and the higher order encourage the use of small functions that factor out duplicated code. As a result, amounts of similar code that wouldn’t raise an eyebrow in the imperative style are extracted and reused in FP style.

The result of applying this principles using my last toy language, Scala, follow this principles.

As you can see, looping has been replaced by the higher-order function Seq.tabulate that accept a function of type Int => T to fill a sequence, in this case the sequence of tree lines. This design can be easily upgraded to generate lazy or even infinite trees by changing Seq by Stream.

Other important point is than several functions and values have been extracted to modularize the tree generation logic. This is easier to extend/refine with more details.

And last but not least, a test suite is included since Tree(n) is very easy to test anyway.

Some of this characteristics can be observed on different submissions you can find on the book: referential transparency on another Scala version by Juan de Bravo and an R version by Arturo Canales, and higher order with Haskell by Ignacio Blasco and Clojure by me.