So to answer your question, brave products including brave search runs on AWS cloud servers and not on their servers.
AWS is already leading in green electricity consumption https://sustainability.aboutamazon.com/environment/the-cloud/cloud-efficiency. So, it is better than brave search from ecological point of view is running on AWS rather than their own servers. You will need to ask official brave team to supply you data from https://aws.amazon.com/blogs/aws/new-customer-carbon-footprint-tool/
The problem with current Ecosia set-up is that are only able to plant trees when you click on their ads. These ads are actually bing ads as they are bing partners. These ads might have some minor non-privacy attribution protocol. Let’s say you are using brave browser with shields set-up on aggressive, then automatically first party ecosia search ads will be blocked by default.
Also, is this you https://libreddit.mha.fi/r/brave_browser/comments/vmjchl/does_anyone_know_if_brave_has_an_environmental/