I think there are two issues here one of which is and has been reasonably explained - not all woodlands are shown due to the scale of the map. It seems that only the largest/densest/ most significant tracts of trees are shown. On the main map for examply Woody End is missing as are the Firienwood and Druadan Forest which do appear in the enlarged map of Rohan, Gondor and Mordor. Barbara Strachey's wonderful maps pick up on every detail and makes educated guesses. I am sure it was a question of space. We know that Ithilien had lovely woods and the Shire I imagine had a similar landscape to the shires Tolkien knew. Cultivated land with hedgerows and copses between fields ane hursts of trees in some of them. Not densely forested but very tree-y.
The other issue which I am spectacularly unqualified to answer having dropped Geography at 13 (which was a little while ago
and we only did Glaciated Highlands and the Borneo Sumba anyway) is how land which which was heavily forested has become not so and remained so despite the absence of population and agriculture or large herds of ruminant animals. I know that the Australian rainforest actually needs the occasional forest fire to regenerate so even if Sauron's force burned the lands of Eriador when Ost in Edhil was destroyed and Rivendell besieges I can't quite see why the trees wouldn't have made more of a return given that the climate wasn't hostile.