Improve speed in this tree - memoization won't suffice

I found that it is considerably faster, if you process the fractional part of j as a seperate integer variable:

Clear[function1, function2]

function1[i_, j_, jc_] := function1[i, j, jc] = 
  Which[i >= 40, 1.0, j >= 40, 0.0, jc >= 40, 0.0, True, 
    0.18*0.45*(1 - function2[j, i + 1, jc]) + 
    0.66*0.45*(1 - function2[j, i + 2, jc]) + 
    0.16*0.45*(1 - function2[j, i + 3, jc]) + (1 - 0.18*0.45 - 
    0.66*0.45 - 0.16*0.45)*(1 - function2[j, i, jc])];


function2[j_, i_, jc_] := function2[j, i, jc] = 
  Which[j >= 40, 1.0, i >= 40, 0.0, True, 
    0.18*0.55*(1 - function1[i, j + 1, jc]) + 
    0.66*0.55*(1 - function1[i, j + 2, jc]) + 
    0.16*0.55*(1 - function1[i, j + 3, jc]) + (1 - 0.18*0.55 - 
    0.66*0.55 - 0.16*0.55)*(1 - function1[i, j, jc + 1])];

The primary inefficiency of your code comes from quite a lot of unnecessary evaluations and memoizations because of pattern matching on machine numbers. If we eliminate these the function is several times faster. Simply changing function1[i, j+0.01] in your last line of code to function1[i, j + 1/100] is sufficient.

With your original:

function1[0, 0] // AbsoluteTiming
function1 // DownValues // Length
function2 // DownValues // Length

{6.90192, 0.218588}

342540

323897

With that sole change:

function1[0, 0] // AbsoluteTiming
function1 // DownValues // Length
function2 // DownValues // Length

{1.48465, 0.218587}

69036

67436

Berg's code fundamentally works in the same way.