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I visited this area in the northern summer of 2019.

These gardens were the first created in Bhutan, and they were opened in 2008. Inside the main accessible central area at the gates, there is the Rhododendron display, a lake, lots of large trees, flowers, and walking tracks, over an area of 50 hectares. However, the park actually extends for another 120 square kilometres between surrounding national parks, and includes the viewpoint and stupas at Dochula Pass.

As with a lot of nature watching in Bhutan, it is often best to simply walk along the roads that wind their way through the mountains. The road leading down to and around the gardens is good for birding.
 Wedge-tailed Green Pigeons
Wedge-tailed Green Pigeons Yellow-billed Blue Magpies
Yellow-billed Blue Magpies Brown Parrotbill
Brown Parrotbill Whiskered Yuhina
Whiskered Yuhina Green-backed Tit
Green-backed Tit Rufous Sibia
Rufous Sibia Striated Bulbul
Striated Bulbul Chestnut Crowned Laughing Thrush
Chestnut Crowned Laughing Thrush White-throated Laughing Thrush
White-throated Laughing Thrush Green-tailed Sunbird
Green-tailed SunbirdYou might also spot the occasional Pika in the undergrowth...

For more places to see wildlife in Bhutan.
For some higher altitude (3988) flowers and birds, head to Chelila Pass. For lower altitude (1500 metres) head to Jigme Dorji National Park.