Conceptually

Skill tree is the mix of progressive mechanics and emergence mechanics.

As Progressive Mechanics

It is commonly believe that skill tree is a handy tool for adjusting in-game balance by game designers.

According to Wikipedia:

Designer can use it to delay player’s acquisition so they are not overwhelmed at the start. It is great for easing in new gameplay dynamics by starting players with scaled down versions, which can be fully unlocked over time. Therefore, designer can control the game flow in an enjoyable way.

Berry gave an excellent explanation of the use of skill tree mechanics in the game (2013):

There's no better way to educate your player in what they can (eventually) do in your game than by having a skill tree that they'll be looking for time and time again thinking about where they want to advance their character. This personalization also feeds a common need for self-expression in play that games often lack, and having a potent and varied talent system allows you to turn your single game's worth of content into an endless sandbox of gameplay discovery.

As Emergence Mechanics

Conversely, skill tree is also emergence mechanics. It has the ability to give player wide possibility space.  Skill tree offers an engaging gameplay in which players’ decisions on how they allocate points into their skill trees will affect their overall gaming experience as they play on. Without it, player will gain a monotonous experience because of the feeling of controlled by the progressive mechanics.

Again Berry provided a good example (2013):

A relatively simple game with the right talent tree can be played indefinitely. If you died in a game that lacks progression aspects it was simply Game Over, but now at least you get a little bit of progress towards that next cool talent that can help you overcome the obstacle that killed you.


Finally, although there are no set rules for designing a skill tree, by its nature, skills are usually arranged into arrays that start small, and then grow into a more complicated form.
These skills are usually organized into different categories by using “Nodes” (which is shown in the image below)

(image from article Let’s Spec Into Talent Trees: A Primer for Game Designers)

While designing skill trees, the design goals, the target audience demand should be our guidelines.
These are the core concepts we as designers should always keep in mind when designing a skill tree that can best serve the game.
     

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"Let’s Spec Into Talent Trees: A Primer for Game Designers". Game Development at Tits plus. Retrieved 01 Feb 2015.