The Barbican has two main gardens, Thomas More Garden and Speed Garden, which both abut the lake. Speed Garden, which is the smaller of the two, is at the eastern end of the lake, between Speed House and Brandon Mews.
The large tree is an Ash; the tree with the white trunk is a Eucalyptus. There used to be many more trees in Speed Garden but the shade was killing the grass so some were cut down.
In the colonnade between Speed Gardens and the lake, the big vine-like bush is a Kiwi Fruit (excellent with German rye bread according to one of the gardeners).
The grass is ordinary grass. The City contracts out the grass cutting. The grass has to be kept no less than 10 cm and no more than 20 cm high, which means it is cut once a week in the Summer.
Ten years ago, they put 25,000 crocuses into the lawns of the two gardens. I imagine they have mainly been eaten by the birds.