I actually had to do something similar to this a couple of days ago.  I found it easiest to do the calculations in Excel and then create an XY Events table in ArcGIS.  I'm not familiar with QGIS but here is how you can do the calculations in Excel.

I'm assuming that there will be two columns in the spreadsheet with the following data

 - Column A: Compass Bearing in decimal degrees
 - Column B: Distance from base point in
   metres

You will then need to create two calculated columns (note these formalae are in Excel format) 

 - Column C: =SIN(RADIANS( [Column A]))*[Column B] 
 - Column D:
   =COS(RADIANS([Column A]))*[Column B]

Column C will contain the X displacement in metres from your base point and Column D will contain the Y displacement in metres from your base point.  If you add the value in Column C to the X coordinate of your base point and the value in Column D to the Y coordinate of your base point you will have the coordinate of the tree.