Cayley Graph of F_2 with subgroups emphasized

Question

I'd like to draw the Cayley graph of F_2 (the free group on two generators), with a special consideration. Namely, that the index 2 subgroups (the vertices reached with an even number of movements and vertices reached with an even number of vertical/horizontal movements) are to be coloured differently. I found this wonderful post Cayley Graph of Free Group in TikZ about creating the graph, though none seemed easily amenable to what I wanted to do. Could someone please help?

Answer 1

This code is built from an example showcasing tikzmath

The result

The code

\documentclass[tikz,border=10pt]{standalone} %built from http://latex-community.org/know-how/513-tikz-math
\usetikzlibrary{math,backgrounds}
\begin{document}

% -------------------
% styles of the beads
% -------------------
\tikzset
{
  coloredBeadRoot/.style=
  {
    circle,
  },
  coloredBead/.style=
  {
    coloredBeadRoot,
    fill=\color,
    scale=\beadsize*\scale,
  }
}

\begin{tikzpicture}[scale=.3pt]
  \tikzmath{
    % --------------------------
    % the parameters of the tree
    % --------------------------
    \power=2.3; % the scale base factor
    \deviation=90; % the angle between the 3 child edges
    \numsteps=4; % number of levels
    let \startcolor=black; % the start color
    let \endcolor=black; % the end color
    let \oddcolor=red; % the odd bead color
    let \evencolor=blue; % the even bead color
    \beadsize=.2; % the scale of colored beads
    % -------------------------------------------------------------------------
    % the function that draw one edge and call itself to draw the 3 child edges
    % -------------------------------------------------------------------------
    function Branch(\x,\y,\rotate,\step){
      \step=\step-1; % stops drawing if step < 0
      if (\step >= 0) then %
      {
        \mix = int(100*\step/(\numsteps-1)); % the color mix parameter is in [0,100]
        \scale = \power^\step; % the scale factor
        { % "print" the tikz command that draw the edge
          \scoped[on background layer]
          \draw
          [
            shift={(\x pt,\y pt)},
            rotate=\rotate,
            scale=\scale,
            color=\startcolor!\mix!\endcolor,
            line width=\scale*.1 pt,
            line cap=round,
          ]
            (0,0)--(1,0) coordinate(newbase) ;
        };
        % -----------------------
        % place the colored beads
        % -----------------------
        if (isodd(\step)) then %
        {
          let \color=\oddcolor;
        }
        else
        {
          let \color=\evencolor;
        };
        {
          \node [coloredBead] at (newbase) {} ;
        };
        % --------------
        % recursive call
        % --------------
        coordinate \b;
        \b1 = (newbase); % the new base point
        for \a in {-\deviation,0,\deviation}%
        {
          Branch(\bx1,\by1,mod(\rotate+\a,360),\step); % draw one child edge
        };
      };
    };
    % --------------
    % draw root bead
    % --------------
    {
      \node 
      [
        coloredBeadRoot,
        fill=\evencolor,
        scale=\beadsize*\power^\numsteps,
      ] 
      at (0,0) {};
    };
    % ----------------------
    % draw the four branches
    % ----------------------
    for \angle in {0,90,180,-90}{
      Branch(0,0,\angle,\numsteps);
    };
  }
\end{tikzpicture}
\end{document}

Cheers