How can I "unlock" the next skill in the tree without perfecting its parent?
From looking around at others' profiles, I see that they've proceeded past a skill despite not perfecting it. In my tree, the next skill is only unlocked upon completion of all lessons in the preceding skill.
How did they do it?
It sounds like you're looking at word strength, which is shown by the golden strength bar on a skill. The checkmark next to the lessons in the skill and the cup signify that you've learned the skill. Over time, your strength in what you've learned will decay. That's why you'll often see that the bar is not full. It means it's time to refresh by doing some practice.
Ah, thanks for clarifying! So, we made updates recently to the Portuguese tree. You can hear about them here: http://duolingo.com/#/comment/402624 In order to alter the tree (add or remove lessons and words) you'll sometimes see that a skill previously learned is no longer complete. That explains what is happening in Wataya's tree.
I had this happen to me when a skill opened up in the Spanish tree where the parent tree was not mastered.. For example when I finished "Adj" , Adverbs and VPres 2 were suppose to open up and they did, but the "object" skill opened up also with one lesson completed. I believe this happened because I learned enough of the words in "Objects" through other lessons, to have competed one lesson in "Objects"..