Interesting name. If you put Ásgarðr through the mill of (northern)
English sound changes, I suppose it would come out as Aysgarth, but
older spellings of the name are Echescard and Aykescarth so probably
oak trees after all, and not Odin's Yorkshire residence...

http://www.genuki.org.uk/big/eng/YKS/NRY/Aysgarth/Aysgarth90.html

ON 'garðr' was borrowed into English where it appears as 'garth', as
you say Patricia. The native cognate is 'yard', from Old English
'geard'. 'garden' comes from Old Norman French 'gardin' from Popular
Latin *gardinum, from a continental Germanic cognate.