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    <p>A tail recursive function is a function where the only recursive call is the last one in the function. A non-tail recursive function is a function where that is not the case.</p> <p>A backward recursion is a recursion where in each recursive call the value of the parameter is less than in the previous step. A forward recursion is a recursion where it grows bigger with each step.</p> <p>Those are two orthogonal concepts, i.e. a forward recursion may or may not be tail-recursive and the same applies to backward recursions.</p> <p>For example the factorial function is often written like this in imperative languages:</p> <pre><code>fac = 1 for i from 1 to n: fac := fac * i </code></pre> <p>The common recursive version of factorial counts backwards (i.e. it calls itself with <code>n-1</code> as the parameter), however if you'd directly translate the above imperative solution, you'd come up with a recursive version that counts upwards. It would look something like this:</p> <pre><code>let fac n = let rec loop i = if i &gt;= n then i else i * loop (i+1) in loop 1 </code></pre> <p>This is a forward recursion and as you can see it is slightly more cumbersome than the backward recursive variant as it requires a helper function. Now this is not tail recursive as the last call in <code>loop</code> is the multiplication, not the recursion. So to make it tail-recursive, you'd do something like this:</p> <pre><code>let fac n = let rec loop acc i = if i &gt;= n then acc else loop (i*acc) (i+1) in loop 1 1 </code></pre> <p>Now this is both a forward recursion and a tail recursion because the recursive call is a) a tail-call and b) calls itself with a greater value (<code>i+1</code>).</p>
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