Scanning basic blocks in reverse is how you calculate liveness. This is ancient, documentedin texts like the Dragon Book, 1988 (from whose perspective it is already old hat).

Liveness extends from definition to use, so you can think about it in the ordinary forward direction. Like when you read code in a high level language, you see some variable definitions near the top of the scope and then look for uses below.

But one problem is that scanning in the forward direction, you don't know that a definition is actually used. A definition of the temporary isn't what makes it live; the uses of the definition is what makes it live. Liveness flows backwards from uses to definitions. The uses say "looking backwards, there is the definition I'm using."

Another problem is, where does the lifetime end? It ends with the last use. Scanning forward, we don't know whether any given use we have seen so far is the last use.

In reverse, the events are in the right order. See a use, the temporary is live there and in prior instructions. If it was live already, nothing changes. See a definition, and it is dead prior to that definition.

This is directly relevant to assigning the temporaries to a smaller set of physical registers.

For that matter, you can use a backward scan to simply reduce the temporaries to a minimal set. So that is to say, take some code that keeps using new temporaries throughout, and do this backward allocation algorithm, using unlimited temporaries (no need to spill). You end up with a version of the code using a smaller set of temporaries.

t1 = a(b, c)
t2 = d(t1)
t3 = e(t1)
t4 = f(t2)
t5 = g(t4)

We want to reduce the t's, but are not constrained to a fixed set of machine registers; we can allocate as many t's as we want. So working backwards:

t5 = g(t4)

t5 is defined. It has no next use; we have not seen it in our backward scan. So we assign it to t'1 and immediately free t'1, so our free stack holds (t'1). t4 is a use; that is live: we assign it to t4 == t'2.
t'2 is live, t'1 isn't:

t'1 = g(t'2)

now we see t4 = f(f2). That means t4 is before this point, which means t'2 is dead. We push t'2 onto our free stack so it holds (t'2 t'1). t2 is newly live. What can we map it to? Why the t'2 we just freed. So t2 == t'2. We generate

t'2 = f(t'2)
t'1 = g(t'2)

Next we have t3 = e(t1). t3 isn't in the use list (hasn't been seen before) and is here dying. We can allocate it to the t'1 we we freed earlier, emptying our free stack to (). Then since it is dead, we immediately free it, so the free stack is (t'1). The used t1 is new, and we allocate it t'1, emptying the free stack:

t'1 = e(t'1)
t'2 = f(t'2)
t'1 = g(t'2)

Next is t2 = d(t1). Aha, here t1 is still live which we mapped to t'1. And t2 is being defined; we have that live as t'2. t'2 is freed now making our free stack (t'2)"

t'2 = d(t'1)
t'1 = e(t'1)
t'2 = f(t'2)
t'1 = g(t'2)

Finally t1 = a(b, c) is where t1 is defined, mapped to t'1. t'1 goes dead so our free stack is (t'1 t'2).

t'1 = a(b, c)
t'2 = d(t'1)
t'1 = e(t'1)
t'2 = f(t'2)
t'1 = g(t'2)

Now we drop the apostrophe:

t1 = a(b, c)
t2 = d(t1)
t1 = e(t1)
t2 = f(t2)
t1 = g(t2)

Look, everything is done with just two temporaries now instead of 5.