Generalize Tournament Bracket Into Valid Team Rankings Looking at a bracket for a single elimination 16 player tournament (like below), the goal is to generate unique orderings from 1 to 16 of the players' skill. 
See this example bracket: Arbitrary Bracket
In this bracket example, we can see that Ernie beat Robert in the finals, therefore we can include 'Ernie> Robert' as a rule when we make our list. Going down the bracket we see that Robert beat Andrew who beat Ian who beat Johan. Therefore, via transitive property we can make a longer rule to follow: 'Ernie> Robert> Andrew> Ian> Johan. Once we lay out all of the rules of how each player must be ordered, we can generate a list from best player to worst player (1 to 16) that satisfies those rules.
I know there are many different possible orderings of the players because, for example, Adrian never plays against Scott, so as long as Scott comes after Lisa and Adrian comes after Kevin, the list is valid regardless of the actual ordering of Adrian and Scott. 
My question is: is there a decent way to generalize a list of rules to apply to a tournament of n (even) number of players (something capable of being efficiently produced with a program)? And how many unique valid orderings are possible for a tournament of n possible players?
 A: I don't understand what you mean by your first question, but I find your second question interesting.
I also don't know how you intend to set up a single elimination tournament with a general even number $n$ of players, so I'll answer the question for the usual case $n=2^k$.
The player who wins the tournament is at the top of the order, so we can disregard them. The rest of the players form subtrees $T_i$ with $2^{i}$ players, with $i=0,\ldots,k-1$, with the roots of the trees given by the winner's victories. For instance, in your example, Drew was beaten by the winner Erwin and thus forms the tree $T_0$; Bert and Zaphod are in the tree $T_1$ with its root at Erwin's victory over Zaphod; Adrian, Lisa, Kevin and Scott are in the tree $T_2$ with its root at Erwin's victory over Lisa; and the remaining eight players are in the tree $T_3$ with its root at Erwin's victory over Robert.
Each tree forms a subtournament, and the orders within the subtournaments are independent of each other and are free to be any of the orders of a tournament of that size. So we can compute the number $C_k$ of possible orders recursively:
$$
C_k=\frac{\left(2^k-1\right)!}{2^0!\cdots2^{k-1}!}\prod_{i=0}^{k-1}C_i\;,
$$
where the multinomial coefficient in front counts the number of ways of choosing slots in the order for the trees, $2^i$ for tree $T_i$ out of a total of $2^k-1$ free slots (since the winner is fixed). The first few values are $C_0=1$, $C_1=1$, 
$$
C_2=\frac{\left(2^2-1\right)!}{2^0!2^1!}=3\;,
$$
$$
C_3=\frac{\left(2^3-1\right)!}{2^0!2^1!2^2!}\cdot3=315\;,
$$
$$
C_4=\frac{\left(2^4-1\right)!}{2^0!2^1!2^2!2^3!}\cdot3\cdot315=1206789333750\;.
$$
That last one is the number of possible orders in your example tournament.
