For the first sight it looks very easy to create the table of standings with League Results "Generate Table & Positions" feature.
And most of the cases it is. Here I will reveal some details helping you if something goes wrong.

At last I created another option making life with LR easier: copying of table contents.
Before version 0.6.2 to create the table after a round of matches you needed many steps including creating the table header and then all positions with teams of the season.

The data structures needed to display properly elimination tree of the tournament are a bit complex. Before the feature of automatic fixture creation is available here is how to set it right and get the view like here on LR Demo page.

As it states in the main description of League Results component it is very easy to modify to meet any special requests, because it is made with MVC architecture. Here is how to create your own layout of League Results component view. I believe it is true for all Joomla 1.5 MVC components.
You need to know at least basics of HTML and PHP to use this document.